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+1,3 @@ +## 欢迎来到jy-hf的代码空间 + +本学习网站使用mkdocs-material搭建，建站教程请见 [mkdocs.org](https://www.mkdocs.org). \ No newline at end of file diff --git a/mkdocs.yml b/mkdocs.yml index 459d9e8b..c061724a 100644 --- a/mkdocs.yml +++ b/mkdocs.yml @@ -87,6 +87,7 @@ markdown_extensions: - pymdownx.keys - pymdownx.superfences - pymdownx.tasklist + - markdown.extensions.extra #python markdown 无法显示内联公式，需要使用pymdownx.arithmatex - pymdownx.arithmatex: generic: true diff --git "a/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/mynotes/FDS-notes/index.html" "b/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/mynotes/FDS-notes/index.html" index e4cb5f13..fe1fac86 100644 --- "a/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/mynotes/FDS-notes/index.html" +++ "b/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/mynotes/FDS-notes/index.html" @@ -1380,150 +1380,153 @@

    树

    中序 12 11 20 17 1 15 8 5

    前序 1 11 12 17 20 5 8 15

    后序 12 20 17 11 15 8 5 1

    -
    level-order(Tree T){
-    //注意level-order使用了while，是enqueue
-    enqueue(T)
-    while(queue is not empty){
-        print(T = dequeue())
-        enqueue(T->left)
-        enaueue(T->right)
-    }
-}
-
-//以下为错误的层序遍历做法！
-void zigzag_levelorder(tree_ptr tree, int rev, queue_ptr queue)
-{
-    enqueue(tree, queue);
-    while (is_empty(queue) == 0)
-    {
-        tree_ptr T = dequeue(queue);
-        visit(T); // 打印出这层的节点
-        // 在此处调整顺序
-        if (rev == 0)
-        {
-            rev = 1;
-            zigzag_levelorder(T->left, rev, queue);//应该改为enqueue~~
-            zigzag_levelorder(T->right, rev, queue);
-        }
-        else
-        {
-            rev = 0;
-            zigzag_levelorder(T->right, rev, queue);
-            zigzag_levelorder(T->left, rev, queue);
-        }
-    }
-}
-//zigzag-levelorder hw4真是多金
-int zigzag_levelorder(tree_ptr tree, queue_ptr queue)
-{
-    // zigzag的层序遍历
-    int level = 0;
-    enqueue(tree, queue);
-    while (is_empty(queue) == 0)
-    {
-        int len = queue->rear - queue->front;
-        if (level % 2)
-        {
-            for (int i = queue->front; i < queue->front + len; i++)
-            {
-                printf(" %d", queue->list[i]->data);
-            }
-        }
-        else
-        {
-            for (int i = queue->front + len - 1; i >= queue->front; i--)
-            {
-                if (level == 0)
-                    printf("%d", queue->list[i]->data);
-                else
-                    printf(" %d", queue->list[i]->data);
-            }
-        }
-        //到此为止，往上：打印，也就是visit
-        //往下，enqueue
-        for (int i = queue->front; i < queue->front + len; i++)
-        {
-            if (queue->list[i]->left != NULL)
-                enqueue(queue->list[i]->left, queue);
-            if (queue->list[i]->right != NULL)
-                enqueue(queue->list[i]->right, queue);
-        }
-        queue->front = queue->front + len;
-        level++;
-    }
-}
-
-
-
-in-order(Tree T){
-    if(T){
-        pre-order(T->left)
-        print(T);
-        pre-order(T->right)
-    }
-}
-
-in-order-iterative(Tree_ptr Tree){
-    //写成iterative的形式,使用栈模拟
-    //先将最左边一条全部进栈，弹出栈顶节点（离叶子层较近的节点）后右节点进栈
-    stack S = init_stack();
-    while(1){
-        while(Tree->left!=NULL)S.push(Tree = Tree->left);
-        Tree = S.pop();
-        if(Tree == NULL)break; //直到stack弹空，退出while循环
-        print(Tree->data);
-        Tree = Tree->right;
-    }
-
-}
-
-pre-order(Tree T){
-    if(T){
-        print(T);
-        pre-order(T->left)
-        pre-order(T->right)
-    }
-}
-
-post-order(Tree T){
-    if(T){
-        post-order(T->left)
-        post-order(T->right)
-        print(T);
-    }
-}
-
    +
    level-order(Tree T){
+    //注意level-order使用了while，是enqueue
+    enqueue(T)
+    while(queue is not empty){
+        print(T = dequeue())
+        enqueue(T->left)
+        enaueue(T->right)
+    }
+}
+
+//以下为错误的层序遍历做法！
+void zigzag_levelorder(tree_ptr tree, int rev, queue_ptr queue)
+{
+    enqueue(tree, queue);
+    while (is_empty(queue) == 0)
+    {
+        tree_ptr T = dequeue(queue);
+        visit(T); // 打印出这层的节点
+        // 在此处调整顺序
+        if (rev == 0)
+        {
+            rev = 1;
+            zigzag_levelorder(T->left, rev, queue);//应该改为enqueue~~
+            zigzag_levelorder(T->right, rev, queue);
+        }
+        else
+        {
+            rev = 0;
+            zigzag_levelorder(T->right, rev, queue);
+            zigzag_levelorder(T->left, rev, queue);
+        }
+    }
+}
+//zigzag-levelorder hw4真是多金
+int zigzag_levelorder(tree_ptr tree, queue_ptr queue)
+{
+    // zigzag的层序遍历
+    int level = 0;
+    enqueue(tree, queue);
+    while (is_empty(queue) == 0)
+    {
+        int len = queue->rear - queue->front;
+        if (level % 2)
+        {
+            for (int i = queue->front; i < queue->front + len; i++)
+            {
+                printf(" %d", queue->list[i]->data);
+            }
+        }
+        else
+        {
+            for (int i = queue->front + len - 1; i >= queue->front; i--)
+            {
+                if (level == 0)
+                    printf("%d", queue->list[i]->data);
+                else
+                    printf(" %d", queue->list[i]->data);
+            }
+        }
+        //到此为止，往上：打印，也就是visit
+        //往下，enqueue
+        for (int i = queue->front; i < queue->front + len; i++)
+        {
+            if (queue->list[i]->left != NULL)
+                enqueue(queue->list[i]->left, queue);
+            if (queue->list[i]->right != NULL)
+                enqueue(queue->list[i]->right, queue);
+        }
+        queue->front = queue->front + len;
+        level++;
+    }
+}
+
+
+
+in-order(Tree T){
+    if(T){
+        pre-order(T->left)
+        print(T);
+        pre-order(T->right)
+    }
+}
+
+in-order-iterative(Tree_ptr Tree){
+    //写成iterative的形式,使用栈模拟
+    //先将最左边一条全部进栈，弹出栈顶节点（离叶子层较近的节点）后右节点进栈
+    stack S = init_stack();
+    while(1){
+        while(Tree->left!=NULL)S.push(Tree = Tree->left);
+        Tree = S.pop();
+        if(Tree == NULL)break; //直到stack弹空，退出while循环
+        print(Tree->data);
+        Tree = Tree->right;
+    }
+
+}
+
+pre-order(Tree T){
+    if(T){
+        print(T);
+        pre-order(T->left)
+        pre-order(T->right)
+    }
+}
+
+post-order(Tree T){
+    if(T){
+        post-order(T->left)
+        post-order(T->right)
+        print(T);
+    }
+}
+
+
+
+

    前序的第一个元素和后序的最后一个元素为根节点

    中序从左到右，实际上的根节点将数组分成了完整的两个部分，因此只要确定当前的根节点，就能找出左、右子树的区间

    中序遍历+后序遍历建树：week-4-hw

    -
    //由中序遍历和后序遍历建立上面的树：
-//中序遍历：12 11 20 17 1 15 8 5 
-//后序遍历：12 20 17 11 15 8 5 1
-//index =             4   
-//11为新树的根节点
-
-tree_ptr build_tree(int *in_order, int *post_order, int n)
-{
-    if (n <= 0 || in_order == NULL || post_order == NULL || n > MAX_NODE_NUM)
-        return NULL;
-    tree_ptr root = (tree_ptr)malloc(sizeof(tree_node));
-    root->data = post_order[n - 1];
-    int index = 0;
-    for (int i = 0; i < n; i++)
-    {
-        if (in_order[i] == post_order[n - 1]) //中+后：取后序区间的最后一个元素，在中序中找到
-            index = i;                        //该元素的index，传入下面的递归
-    }
-    // 把中序序列分成两部分，左边的是左子树，右边的是右子树
-    // 此处 in_order + index + 1 忘记+1，导致只有左枝的左子树建成了右子树
-    root->left = build_tree(in_order, post_order, index);
-    // in_order/post_order的[0~index-1],共index个数
-    root->right = build_tree(in_order + index + 1, post_order + index, n - index - 1);
-    // in_order/post_order的[index+1~n-1]，共n-1-(index+1)+1 = n-1-index个数
-    //左右加起来为n-1个数，刚好少了根节点
-    return root;
-}
-
    +
    //由中序遍历和后序遍历建立上面的树：
+//中序遍历：12 11 20 17 1 15 8 5 
+//后序遍历：12 20 17 11 15 8 5 1
+//index =             4   
+//11为新树的根节点
+
+tree_ptr build_tree(int *in_order, int *post_order, int n)
+{
+    if (n <= 0 || in_order == NULL || post_order == NULL || n > MAX_NODE_NUM)
+        return NULL;
+    tree_ptr root = (tree_ptr)malloc(sizeof(tree_node));
+    root->data = post_order[n - 1];
+    int index = 0;
+    for (int i = 0; i < n; i++)
+    {
+        if (in_order[i] == post_order[n - 1]) //中+后：取后序区间的最后一个元素，在中序中找到
+            index = i;                        //该元素的index，传入下面的递归
+    }
+    // 把中序序列分成两部分，左边的是左子树，右边的是右子树
+    // 此处 in_order + index + 1 忘记+1，导致只有左枝的左子树建成了右子树
+    root->left = build_tree(in_order, post_order, index);
+    // in_order/post_order的[0~index-1],共index个数
+    root->right = build_tree(in_order + index + 1, post_order + index, n - index - 1);
+    // in_order/post_order的[index+1~n-1]，共n-1-(index+1)+1 = n-1-index个数
+    //左右加起来为n-1个数，刚好少了根节点
+    return root;
+}
+

    形象一点：一遍过后：

    中序 （左子树：12 11 20 17 ）【1】 （右子树：15 8 5）

    后序 （12 20 17 11）（ 15 8 5）【1】

    @@ -1531,633 +1534,639 @@

    中序遍历+后序遍历建树：week-4-hw

    中序 （（12） 11 （20 17） ）【1】 （（15 8 ）{5}（））

    后序 （（12） （20 17） {11}）（ （15 8） {5}）【1】

    前序遍历+中序遍历建树：lab2

    -
    tree_ptr build_tree(int *pre_order, int pre_left, int pre_right, int *in_order, int in_left, int in_right)
-{
-    // 先序遍历建树,先处理根节点，再处理左子树，再处理右子树
-    if (pre_left > pre_right || in_left > in_right)
-    {
-        return NULL;
-    }
-    int k;
-    for (k = in_left; k <= in_right; k++)
-    {
-        // 找到根节点在中序遍历中的位置
-        if (in_order[k] == abs(pre_order[pre_left]))
-        {
-            break;
-        }
-    }
-    int num_left = k - in_left; // 左子树的节点数
-    tree_ptr root = (tree_ptr)malloc(sizeof(tree_node));
-    root->data = abs(pre_order[pre_left]);
-    root->color = red_or_black(pre_order[pre_left]);
-    root->left = build_tree(pre_order, pre_left + 1, pre_left + num_left, in_order, in_left, k - 1); //除开pre_order[0](根节点),左树的区间[pre_left+1,pre_left+num_left-1]
-    root->right = build_tree(pre_order, pre_left + num_left + 1, pre_right, in_order, k + 1, in_right);
-    return root;
-}
-
    +
    tree_ptr build_tree(int *pre_order, int pre_left, int pre_right, int *in_order, int in_left, int in_right)
+{
+    // 先序遍历建树,先处理根节点，再处理左子树，再处理右子树
+    if (pre_left > pre_right || in_left > in_right)
+    {
+        return NULL;
+    }
+    int k;
+    for (k = in_left; k <= in_right; k++)
+    {
+        // 找到根节点在中序遍历中的位置
+        if (in_order[k] == abs(pre_order[pre_left]))
+        {
+            break;
+        }
+    }
+    int num_left = k - in_left; // 左子树的节点数
+    tree_ptr root = (tree_ptr)malloc(sizeof(tree_node));
+    root->data = abs(pre_order[pre_left]);
+    root->color = red_or_black(pre_order[pre_left]);
+    root->left = build_tree(pre_order, pre_left + 1, pre_left + num_left, in_order, in_left, k - 1); //除开pre_order[0](根节点),左树的区间[pre_left+1,pre_left+num_left-1]
+    root->right = build_tree(pre_order, pre_left + num_left + 1, pre_right, in_order, k + 1, in_right);
+    return root;
+}
+

    中序 （12 11 20 17） 【1】 （15 8 5）

    前序【1】 （11 12 17 20）（ 5 8 15）

    -
    //自己写写 BST ADT，看看有什么问题
-- SearchTree  MakeEmpty( SearchTree T ){
-    T->left = NULL;
-    T->right = NULL;
-    T->
-  }
-- Position  Find( ElementType X, SearchTree T ){
-    //查找元素，从T开始找，返回指向该节点的position，一个node*
-    if(T==NULL)return NULL;//没有找到
-    if(T->data == X)return T;
-    if(T->->data< X){
-        Find(T->right);
-    }
-    else if(T->data > X){
-        Find(T->left);
-    }
-
-}
-
-- Position  FindMin( SearchTree T ){
-    //找最小元素
-    if(T==NULL)return NULL;//当前树不存在
-    else{
-        if(T->left!=NULL){
-            FindMin(T->left)
-        }
-    }
-    return T;
-    /*别用while:while(T->left!=NULL){
-        T = T->left;
-        FindMin(T);
-    }
-    return T;*/
-}
-- Position  FindMax( SearchTree T ){
-    //同上
-    while(T->right!=NULL){
-        T = T->right;
-        FindMax(T);
-    }
-    return T;
-}
-- SearchTree  Insert( ElementType X, SearchTree T ){
-    //二叉树的正确插入顺序
-    if(T==NULL){
-        node* new_node = (node*)malloc(sizeof(node));
-        T->data = X;
-        T->left = NULL;
-        T->right = NULL;
-        return T;
-    }
-    if(T->data>X){
-        Insert(X,T->left);
-    }
-    else if(T->data<X){
-        Insert(X,T->right);
-    }
-    return T;
-}
-- SearchTree  Delete( ElementType X, SearchTree T ){
-    //删除某元素，是在查找的基础上
-    SearchTree pos = FindMax(X,T);
-    //分类讨论，若pos的度为0/1/2
-    if(pos==NULL)return;
-    else if(pos->left = NULL && pos->right){
-        pos = pos->right
-    }
-    else if(pos->right = NULL && pos->left){
-        pos = pos->left
-    }
-    else{
-        //度为2,用左子树的最大节点替换当前pos指向的根节点
-        SearchTree pos_right_max = FindMax(pos->right)
-        pos->data = pos_right_max->data;
-        Delete(pos->data,pos->right);
-        return pos;
-    }
-}
-- ElementType  Retrieve( Position P ){
-    return P->data;
-}
-
    +
    //自己写写 BST ADT，看看有什么问题
+- SearchTree  MakeEmpty( SearchTree T ){
+    T->left = NULL;
+    T->right = NULL;
+    T->
+  }
+- Position  Find( ElementType X, SearchTree T ){
+    //查找元素，从T开始找，返回指向该节点的position，一个node*
+    if(T==NULL)return NULL;//没有找到
+    if(T->data == X)return T;
+    if(T->->data< X){
+        Find(T->right);
+    }
+    else if(T->data > X){
+        Find(T->left);
+    }
+
+}
+
+- Position  FindMin( SearchTree T ){
+    //找最小元素
+    if(T==NULL)return NULL;//当前树不存在
+    else{
+        if(T->left!=NULL){
+            FindMin(T->left)
+        }
+    }
+    return T;
+    /*别用while:while(T->left!=NULL){
+        T = T->left;
+        FindMin(T);
+    }
+    return T;*/
+}
+- Position  FindMax( SearchTree T ){
+    //同上
+    while(T->right!=NULL){
+        T = T->right;
+        FindMax(T);
+    }
+    return T;
+}
+- SearchTree  Insert( ElementType X, SearchTree T ){
+    //二叉树的正确插入顺序
+    if(T==NULL){
+        node* new_node = (node*)malloc(sizeof(node));
+        T->data = X;
+        T->left = NULL;
+        T->right = NULL;
+        return T;
+    }
+    if(T->data>X){
+        Insert(X,T->left);
+    }
+    else if(T->data<X){
+        Insert(X,T->right);
+    }
+    return T;
+}
+- SearchTree  Delete( ElementType X, SearchTree T ){
+    //删除某元素，是在查找的基础上
+    SearchTree pos = FindMax(X,T);
+    //分类讨论，若pos的度为0/1/2
+    if(pos==NULL)return;
+    else if(pos->left = NULL && pos->right){
+        pos = pos->right
+    }
+    else if(pos->right = NULL && pos->left){
+        pos = pos->left
+    }
+    else{
+        //度为2,用左子树的最大节点替换当前pos指向的根节点
+        SearchTree pos_right_max = FindMax(pos->right)
+        pos->data = pos_right_max->data;
+        Delete(pos->data,pos->right);
+        return pos;
+    }
+}
+- ElementType  Retrieve( Position P ){
+    return P->data;
+}
+
+

    双端队列 double-ended queue，简称Deque

    链表实现

    -
    #include <stdio.h>
-#include <stdlib.h>
-
-// 定义双端队列节点结构
-struct Node {
-    int data;
-    struct Node* next;
-};
-
-// 定义双端队列结构
-struct Deque {
-    struct Node* front;  // 队头指针
-    struct Node* rear;   // 队尾指针
-};
-
-// 创建一个新的双端队列
-struct Deque* createDeque() {
-    struct Deque* deque = (struct Deque*)malloc(sizeof(struct Deque));
-    deque->front = NULL;
-    deque->rear = NULL;
-    return deque;
-}
-//**************************
-// 在队头插入元素，脑袋要有那个三角形，先将新节点指向front指向的node，然后再将front移到新节点上
-//front想象成一个数据域为黑，只有指针的dummy node
-//且总是先将新节点各参数赋值完毕后再进行串联
-//**************************
-void insertFront(struct Deque* deque, int data) {
-    struct Node* newNode = (struct Node*)malloc(sizeof(struct Node));
-    newNode->data = data;
-    newNode->next = deque->front;
-    deque->front = newNode;
-    if (deque->rear == NULL) {
-        deque->rear = newNode;
-    }
-}
-
-// 在队尾插入元素，若rear==NULL 则说明deque为空，此时与front有关，否则只需处理rear指针
-void insertRear(struct Deque* deque, int data) {
-    struct Node* newNode = (struct Node*)malloc(sizeof(struct Node));
-    newNode->data = data;
-    newNode->next = NULL;
-    if (deque->rear == NULL) {
-        deque->front = newNode;
-        deque->rear = newNode;
-    } else {
-        deque->rear->next = newNode;
-        deque->rear = newNode;
-    }
-}
-
-// 从队头删除元素,注意无论是从队头/队尾删除时，先要保证头/尾指针不是NULL，在删除完后还要判断是否删掉了
-//deque中的最后一个元素，如果是则要将front和rear都赋值为NULL
-void deleteFront(struct Deque* deque) {
-    if (deque->front != NULL) {
-        struct Node* temp = deque->front;
-        deque->front = deque->front->next;
-        free(temp);
-        if (deque->front == NULL) {
-            deque->rear = NULL;
-        }
-    }
-}
-
-// 从队尾删除元素
-void deleteRear(struct Deque* deque) {
-    if (deque->rear != NULL) {
-        if (deque->front == deque->rear) {
-            free(deque->front);
-            deque->front = NULL;
-            deque->rear = NULL;
-        } else {
-            struct Node* temp = deque->front;
-            while (temp->next != deque->rear) {
-                temp = temp->next;
-            }
-            free(deque->rear);
-            deque->rear = temp;//temp向后遍历的过程，易出填空题
-            deque->rear->next = NULL;
-        }
-    }
-}
-
-// 检查双端队列是否为空
-int isEmpty(struct Deque* deque) {
-    return (deque->front == NULL);
-}
-
-// 打印双端队列中的元素
-void printDeque(struct Deque* deque) {
-    struct Node* current = deque->front;
-    while (current != NULL) {
-        printf("%d ", current->data);
-        current = current->next;
-    }
-    printf("\n");
-}
-
-// 主函数
-int main() {
-    struct Deque* deque = createDeque();
-
-    insertFront(deque, 1);
-    insertRear(deque, 2);
-    insertFront(deque, 3);
-
-    printDeque(deque);
-
-    deleteFront(deque);
-    printDeque(deque);
-
-    insertRear(deque, 4);
-    printDeque(deque);
-
-    deleteRear(deque);
-    printDeque(deque);
-
-    return 0;
-}
-
    +
    #include <stdio.h>
+#include <stdlib.h>
+
+// 定义双端队列节点结构
+struct Node {
+    int data;
+    struct Node* next;
+};
+
+// 定义双端队列结构
+struct Deque {
+    struct Node* front;  // 队头指针
+    struct Node* rear;   // 队尾指针
+};
+
+// 创建一个新的双端队列
+struct Deque* createDeque() {
+    struct Deque* deque = (struct Deque*)malloc(sizeof(struct Deque));
+    deque->front = NULL;
+    deque->rear = NULL;
+    return deque;
+}
+//**************************
+// 在队头插入元素，脑袋要有那个三角形，先将新节点指向front指向的node，然后再将front移到新节点上
+//front想象成一个数据域为黑，只有指针的dummy node
+//且总是先将新节点各参数赋值完毕后再进行串联
+//**************************
+void insertFront(struct Deque* deque, int data) {
+    struct Node* newNode = (struct Node*)malloc(sizeof(struct Node));
+    newNode->data = data;
+    newNode->next = deque->front;
+    deque->front = newNode;
+    if (deque->rear == NULL) {
+        deque->rear = newNode;
+    }
+}
+
+// 在队尾插入元素，若rear==NULL 则说明deque为空，此时与front有关，否则只需处理rear指针
+void insertRear(struct Deque* deque, int data) {
+    struct Node* newNode = (struct Node*)malloc(sizeof(struct Node));
+    newNode->data = data;
+    newNode->next = NULL;
+    if (deque->rear == NULL) {
+        deque->front = newNode;
+        deque->rear = newNode;
+    } else {
+        deque->rear->next = newNode;
+        deque->rear = newNode;
+    }
+}
+
+// 从队头删除元素,注意无论是从队头/队尾删除时，先要保证头/尾指针不是NULL，在删除完后还要判断是否删掉了
+//deque中的最后一个元素，如果是则要将front和rear都赋值为NULL
+void deleteFront(struct Deque* deque) {
+    if (deque->front != NULL) {
+        struct Node* temp = deque->front;
+        deque->front = deque->front->next;
+        free(temp);
+        if (deque->front == NULL) {
+            deque->rear = NULL;
+        }
+    }
+}
+
+// 从队尾删除元素
+void deleteRear(struct Deque* deque) {
+    if (deque->rear != NULL) {
+        if (deque->front == deque->rear) {
+            free(deque->front);
+            deque->front = NULL;
+            deque->rear = NULL;
+        } else {
+            struct Node* temp = deque->front;
+            while (temp->next != deque->rear) {
+                temp = temp->next;
+            }
+            free(deque->rear);
+            deque->rear = temp;//temp向后遍历的过程，易出填空题
+            deque->rear->next = NULL;
+        }
+    }
+}
+
+// 检查双端队列是否为空
+int isEmpty(struct Deque* deque) {
+    return (deque->front == NULL);
+}
+
+// 打印双端队列中的元素
+void printDeque(struct Deque* deque) {
+    struct Node* current = deque->front;
+    while (current != NULL) {
+        printf("%d ", current->data);
+        current = current->next;
+    }
+    printf("\n");
+}
+
+// 主函数
+int main() {
+    struct Deque* deque = createDeque();
+
+    insertFront(deque, 1);
+    insertRear(deque, 2);
+    insertFront(deque, 3);
+
+    printDeque(deque);
+
+    deleteFront(deque);
+    printDeque(deque);
+
+    insertRear(deque, 4);
+    printDeque(deque);
+
+    deleteRear(deque);
+    printDeque(deque);
+
+    return 0;
+}
+
+

    数组实现

    -
    #include <stdio.h>
-#include <stdlib.h>
-
-#define MAX_SIZE 100
-
-// 定义双端队列结构
-struct Deque {
-    int arr[MAX_SIZE];
-    int front;
-    int rear;
-};
-
-// 创建一个新的双端队列
-struct Deque* createDeque() {
-    struct Deque* deque = (struct Deque*)malloc(sizeof(struct Deque));
-    deque->front = -1;
-    deque->rear = -1;
-    return deque;
-}
-
-// 在队头插入元素
-void insertFront(struct Deque* deque, int data) {
-    if (deque->front == -1) {
-        deque->front = 0;
-        deque->rear = 0;
-        deque->arr[deque->front] = data;
-    } else if (deque->front > 0) {
-        deque->arr[--deque->front] = data;
-    } else {
-        printf("Deque is full (front).\n");
-    }
-}
-
-// 在队尾插入元素
-void insertRear(struct Deque* deque, int data) {
-    if (deque->rear == -1) {
-        deque->front = 0;
-        deque->rear = 0;
-        deque->arr[deque->rear] = data;
-    } else if (deque->rear < MAX_SIZE - 1) {
-        deque->arr[++deque->rear] = data;
-    } else {
-        printf("Deque is full (rear).\n");
-    }
-}
-
-// 从队头删除元素
-void deleteFront(struct Deque* deque) {
-    if (deque->front != -1) {
-        if (deque->front == deque->rear) {
-            deque->front = -1;
-            deque->rear = -1;
-        } else {
-            deque->front++;
-        }
-    } else {
-        printf("Deque is empty (front).\n");
-    }
-}
-
-// 从队尾删除元素
-void deleteRear(struct Deque* deque) {
-    if (deque->rear != -1) {
-        if (deque->front == deque->rear) {
-            deque->front = -1;
-            deque->rear = -1;
-        } else {
-            deque->rear--;
-        }
-    } else {
-        printf("Deque is empty (rear).\n");
-    }
-}
-
-// 检查双端队列是否为空
-int isEmpty(struct Deque* deque) {
-    return (deque->front == -1);
-}
-
-// 打印双端队列中的元素
-void printDeque(struct Deque* deque) {
-    if (isEmpty(deque)) {
-        printf("Deque is empty.\n");
-        return;
-    }
-
-    printf("Front: %d, Rear: %d\n", deque->front, deque->rear);
-    printf("Elements: ");
-    for (int i = deque->front; i <= deque->rear; i++) {
-        printf("%d ", deque->arr[i]);
-    }
-    printf("\n");
-}
-
-// 主函数
-int main() {
-    struct Deque* deque = createDeque();
-
-    insertFront(deque, 1);
-    insertRear(deque, 2);
-    insertFront(deque, 3);
-
-    printDeque(deque);
-
-    deleteFront(deque);
-    printDeque(deque);
-
-    insertRear(deque, 4);
-    printDeque(deque);
-
-    deleteRear(deque);
-    printDeque(deque);
-
    +
    #include <stdio.h>
+#include <stdlib.h>
+
+#define MAX_SIZE 100
+
+// 定义双端队列结构
+struct Deque {
+    int arr[MAX_SIZE];
+    int front;
+    int rear;
+};
+
+// 创建一个新的双端队列
+struct Deque* createDeque() {
+    struct Deque* deque = (struct Deque*)malloc(sizeof(struct Deque));
+    deque->front = -1;
+    deque->rear = -1;
+    return deque;
+}
+
+// 在队头插入元素
+void insertFront(struct Deque* deque, int data) {
+    if (deque->front == -1) {
+        deque->front = 0;
+        deque->rear = 0;
+        deque->arr[deque->front] = data;
+    } else if (deque->front > 0) {
+        deque->arr[--deque->front] = data;
+    } else {
+        printf("Deque is full (front).\n");
+    }
+}
+
+// 在队尾插入元素
+void insertRear(struct Deque* deque, int data) {
+    if (deque->rear == -1) {
+        deque->front = 0;
+        deque->rear = 0;
+        deque->arr[deque->rear] = data;
+    } else if (deque->rear < MAX_SIZE - 1) {
+        deque->arr[++deque->rear] = data;
+    } else {
+        printf("Deque is full (rear).\n");
+    }
+}
+
+// 从队头删除元素
+void deleteFront(struct Deque* deque) {
+    if (deque->front != -1) {
+        if (deque->front == deque->rear) {
+            deque->front = -1;
+            deque->rear = -1;
+        } else {
+            deque->front++;
+        }
+    } else {
+        printf("Deque is empty (front).\n");
+    }
+}
+
+// 从队尾删除元素
+void deleteRear(struct Deque* deque) {
+    if (deque->rear != -1) {
+        if (deque->front == deque->rear) {
+            deque->front = -1;
+            deque->rear = -1;
+        } else {
+            deque->rear--;
+        }
+    } else {
+        printf("Deque is empty (rear).\n");
+    }
+}
+
+// 检查双端队列是否为空
+int isEmpty(struct Deque* deque) {
+    return (deque->front == -1);
+}
+
+// 打印双端队列中的元素
+void printDeque(struct Deque* deque) {
+    if (isEmpty(deque)) {
+        printf("Deque is empty.\n");
+        return;
+    }
+
+    printf("Front: %d, Rear: %d\n", deque->front, deque->rear);
+    printf("Elements: ");
+    for (int i = deque->front; i <= deque->rear; i++) {
+        printf("%d ", deque->arr[i]);
+    }
+    printf("\n");
+}
+
+// 主函数
+int main() {
+    struct Deque* deque = createDeque();
+
+    insertFront(deque, 1);
+    insertRear(deque, 2);
+    insertFront(deque, 3);
+
+    printDeque(deque);
+
+    deleteFront(deque);
+    printDeque(deque);
+
+    insertRear(deque, 4);
+    printDeque(deque);
+
+    deleteRear(deque);
+    printDeque(deque);
+
+
+

    循环队列

    链表实现

    -
    #include <stdio.h>
-#include <stdlib.h>
-
-// 定义循环队列节点结构
-struct Node {
-    int data;
-    struct Node* next;
-};
-
-// 定义循环队列结构
-struct CircularQueue {
-    struct Node* front;
-    struct Node* rear;
-};
-
-// 创建一个新的循环队列
-struct CircularQueue* createCircularQueue() {
-    struct CircularQueue* queue = (struct CircularQueue*)malloc(sizeof(struct CircularQueue));
-    queue->front = NULL;
-    queue->rear = NULL;
-    return queue;
-}
-
-// 入队操作
-void enqueue(struct CircularQueue* queue, int data) {
-    struct Node* newNode = (struct Node*)malloc(sizeof(struct Node));
-    newNode->data = data;
-    newNode->next = NULL;
-    if (queue->rear == NULL) {
-        queue->front = newNode;
-        queue->rear = newNode;
-        newNode->next = newNode; // 链接到自身以构成循环
-    } else {
-        newNode->next = queue->front;
-        queue->rear->next = newNode;
-        queue->rear = newNode;
-    }
-}
-
-// 出队操作
-int dequeue(struct CircularQueue* queue) {
-    if (queue->front == NULL) {
-        printf("Queue is empty.\n");
-        return -1; // 队列为空时返回-1
-    }
-    int data = queue->front->data;
-    struct Node* temp = queue->front;
-    if (queue->front == queue->rear) {
-        queue->front = NULL;
-        queue->rear = NULL;
-    } else {
-        queue->front = queue->front->next;
-        queue->rear->next = queue->front;
-    }
-    free(temp);
-    return data;
-}
-
-// 检查循环队列是否为空
-int isEmpty(struct CircularQueue* queue) {
-    return (queue->front == NULL);
-}
-
-// 打印循环队列中的元素
-void printCircularQueue(struct CircularQueue* queue) {
-    struct Node* current = queue->front;
-    if (current == NULL) {
-        printf("Circular Queue is empty.\n");
-        return;
-    }
-    do {
-        printf("%d ", current->data);
-        current = current->next;
-    } while (current != queue->front);
-    printf("\n");
-}
-
-// 主函数
-int main() {
-    struct CircularQueue* queue = createCircularQueue();
-
-    enqueue(queue, 1);
-    enqueue(queue, 2);
-    enqueue(queue, 3);
-    enqueue(queue, 4);
-
-    printCircularQueue(queue);
-
-    dequeue(queue);
-    dequeue(queue);
-
-    printCircularQueue(queue);
-
-    enqueue(queue, 5);
-    printCircularQueue(queue);
-
-    dequeue(queue);
-    dequeue(queue);
-    dequeue(queue);
-    dequeue(queue);
-
-    printCircularQueue(queue);
-
-    return 0;
-}
-
    +
    #include <stdio.h>
+#include <stdlib.h>
+
+// 定义循环队列节点结构
+struct Node {
+    int data;
+    struct Node* next;
+};
+
+// 定义循环队列结构
+struct CircularQueue {
+    struct Node* front;
+    struct Node* rear;
+};
+
+// 创建一个新的循环队列
+struct CircularQueue* createCircularQueue() {
+    struct CircularQueue* queue = (struct CircularQueue*)malloc(sizeof(struct CircularQueue));
+    queue->front = NULL;
+    queue->rear = NULL;
+    return queue;
+}
+
+// 入队操作
+void enqueue(struct CircularQueue* queue, int data) {
+    struct Node* newNode = (struct Node*)malloc(sizeof(struct Node));
+    newNode->data = data;
+    newNode->next = NULL;
+    if (queue->rear == NULL) {
+        queue->front = newNode;
+        queue->rear = newNode;
+        newNode->next = newNode; // 链接到自身以构成循环
+    } else {
+        newNode->next = queue->front;
+        queue->rear->next = newNode;
+        queue->rear = newNode;
+    }
+}
+
+// 出队操作
+int dequeue(struct CircularQueue* queue) {
+    if (queue->front == NULL) {
+        printf("Queue is empty.\n");
+        return -1; // 队列为空时返回-1
+    }
+    int data = queue->front->data;
+    struct Node* temp = queue->front;
+    if (queue->front == queue->rear) {
+        queue->front = NULL;
+        queue->rear = NULL;
+    } else {
+        queue->front = queue->front->next;
+        queue->rear->next = queue->front;
+    }
+    free(temp);
+    return data;
+}
+
+// 检查循环队列是否为空
+int isEmpty(struct CircularQueue* queue) {
+    return (queue->front == NULL);
+}
+
+// 打印循环队列中的元素
+void printCircularQueue(struct CircularQueue* queue) {
+    struct Node* current = queue->front;
+    if (current == NULL) {
+        printf("Circular Queue is empty.\n");
+        return;
+    }
+    do {
+        printf("%d ", current->data);
+        current = current->next;
+    } while (current != queue->front);
+    printf("\n");
+}
+
+// 主函数
+int main() {
+    struct CircularQueue* queue = createCircularQueue();
+
+    enqueue(queue, 1);
+    enqueue(queue, 2);
+    enqueue(queue, 3);
+    enqueue(queue, 4);
+
+    printCircularQueue(queue);
+
+    dequeue(queue);
+    dequeue(queue);
+
+    printCircularQueue(queue);
+
+    enqueue(queue, 5);
+    printCircularQueue(queue);
+
+    dequeue(queue);
+    dequeue(queue);
+    dequeue(queue);
+    dequeue(queue);
+
+    printCircularQueue(queue);
+
+    return 0;
+}
+
+

    数组实现

    -
    #include <stdio.h>
-#include <stdlib.h>
-
-#define MAX_SIZE 100
-
-// 定义循环队列结构
-struct CircularQueue {
-    int arr[MAX_SIZE];
-    int front;
-    int rear;
-};
-
-// 创建一个新的循环队列
-struct CircularQueue* createCircularQueue() {
-    struct CircularQueue* queue = (struct CircularQueue*)malloc(sizeof(struct CircularQueue));
-    queue->front = -1;
-    queue->rear = -1;
-    return queue;
-}
-
-// 入队操作
-void enqueue(struct CircularQueue* queue, int data) {
-    if ((queue->rear + 1) % MAX_SIZE == queue->front) {
-        printf("Queue is full.\n");
-        return;
-    }
-    if (queue->front == -1) {
-        queue->front = 0;
-        queue->rear = 0;
-        queue->arr[queue->rear] = data;
-    } else {
-        queue->rear = (queue->rear + 1) % MAX_SIZE;
-        queue->arr[queue->rear] = data;
-    }
-}
-
-// 出队操作
-int dequeue(struct CircularQueue* queue) {
-    if (queue->front == -1) {
-        printf("Queue is empty.\n");
-        return -1;
-    }
-    int data = queue->arr[queue->front];
-    if (queue->front == queue->rear) {
-        queue->front = -1;
-        queue->rear = -1;
-    } else {
-        queue->front = (queue->front + 1) % MAX_SIZE;
-    }
-    return data;
-}
-
-// 检查循环队列是否为空
-int isEmpty(struct CircularQueue* queue) {
-    return (queue->front == -1);
-}
-
-// 打印循循环队列中的元素
-void printCircularQueue(struct CircularQueue* queue) {
-    if (isEmpty(queue)) {
-        printf("Circular Queue is empty.\n");
-        return;
-    }
-
-    int i = queue->front;
-    do {
-        printf("%d ", queue->arr[i]);
-        i = (i + 1) % MAX_SIZE;
-    } while (i != (queue->rear + 1) % MAX_SIZE);
-    printf("\n");
-}
-
-// 主函数
-int main() {
-    struct CircularQueue* queue = createCircularQueue();
-
-    enqueue(queue, 1);
-    enqueue(queue, 2);
-    enqueue(queue, 3);
-    enqueue(queue, 4);
-
-    printCircularQueue(queue);
-
-    dequeue(queue);
-    dequeue(queue);
-
-    printCircularQueue(queue);
-
-    enqueue(queue, 5);
-    printCircularQueue(queue);
-
-    dequeue(queue);
-    dequeue(queue);
-    dequeue(queue);
-    dequeue(queue);
-
-    printCircularQueue(queue);
-
-    return 0;
-}
-
    +
    #include <stdio.h>
+#include <stdlib.h>
+
+#define MAX_SIZE 100
+
+// 定义循环队列结构
+struct CircularQueue {
+    int arr[MAX_SIZE];
+    int front;
+    int rear;
+};
+
+// 创建一个新的循环队列
+struct CircularQueue* createCircularQueue() {
+    struct CircularQueue* queue = (struct CircularQueue*)malloc(sizeof(struct CircularQueue));
+    queue->front = -1;
+    queue->rear = -1;
+    return queue;
+}
+
+// 入队操作
+void enqueue(struct CircularQueue* queue, int data) {
+    if ((queue->rear + 1) % MAX_SIZE == queue->front) {
+        printf("Queue is full.\n");
+        return;
+    }
+    if (queue->front == -1) {
+        queue->front = 0;
+        queue->rear = 0;
+        queue->arr[queue->rear] = data;
+    } else {
+        queue->rear = (queue->rear + 1) % MAX_SIZE;
+        queue->arr[queue->rear] = data;
+    }
+}
+
+// 出队操作
+int dequeue(struct CircularQueue* queue) {
+    if (queue->front == -1) {
+        printf("Queue is empty.\n");
+        return -1;
+    }
+    int data = queue->arr[queue->front];
+    if (queue->front == queue->rear) {
+        queue->front = -1;
+        queue->rear = -1;
+    } else {
+        queue->front = (queue->front + 1) % MAX_SIZE;
+    }
+    return data;
+}
+
+// 检查循环队列是否为空
+int isEmpty(struct CircularQueue* queue) {
+    return (queue->front == -1);
+}
+
+// 打印循循环队列中的元素
+void printCircularQueue(struct CircularQueue* queue) {
+    if (isEmpty(queue)) {
+        printf("Circular Queue is empty.\n");
+        return;
+    }
+
+    int i = queue->front;
+    do {
+        printf("%d ", queue->arr[i]);
+        i = (i + 1) % MAX_SIZE;
+    } while (i != (queue->rear + 1) % MAX_SIZE);
+    printf("\n");
+}
+
+// 主函数
+int main() {
+    struct CircularQueue* queue = createCircularQueue();
+
+    enqueue(queue, 1);
+    enqueue(queue, 2);
+    enqueue(queue, 3);
+    enqueue(queue, 4);
+
+    printCircularQueue(queue);
+
+    dequeue(queue);
+    dequeue(queue);
+
+    printCircularQueue(queue);
+
+    enqueue(queue, 5);
+    printCircularQueue(queue);
+
+    dequeue(queue);
+    dequeue(queue);
+    dequeue(queue);
+    dequeue(queue);
+
+    printCircularQueue(queue);
+
+    return 0;
+}
+

    双向循环链表

    -
    #include <stdio.h>
-#include <stdlib.h>
-
-// 定义双向循环链表节点结构
-struct Node {
-    int data;
-    struct Node* next;
-    struct Node* prev;
-};
-
-// 创建一个新的双向循环链表
-struct Node* createDoublyCircularLinkedList(int data) {
-    struct Node* newNode = (struct Node*)malloc(sizeof(struct Node));
-    newNode->data = data;
-    newNode->next = newNode;
-    newNode->prev = newNode;
-    return newNode;
-}
-
-// 在双向循环链表的末尾插入节点
-void insertAtEnd(struct Node** head, int data) {
-    struct Node* newNode = (struct Node*)malloc(sizeof(struct Node));
-    newNode->data = data;
-    newNode->next = (*head);
-    newNode->prev = (*head)->prev;
-    (*head)->prev->next = newNode;
-    (*head)->prev = newNode;
-}
-
-// 打印双向循环链表的元素
-void printDoublyCircularLinkedList(struct Node* head) {
-    if (head == NULL) {
-        printf("Empty Doubly Circular Linked List\n");
-        return;
-    }
-
-    struct Node* current = head;
-    do {
-        printf("%d ", current->data);
-        current = current->next;
-    } while (current != head);
-    printf("\n");
-}
-
-// 主函数
-int main() {
-    struct Node* head = createDoublyCircularLinkedList(1);
-
-    insertAtEnd(&head, 2);
-    insertAtEnd(&head, 3);
-
-    printf("Doubly Circular Linked List: ");c 
-    printDoublyCircularLinkedList(head);
-
-    return 0;
-}
-
    +
    #include <stdio.h>
+#include <stdlib.h>
+
+// 定义双向循环链表节点结构
+struct Node {
+    int data;
+    struct Node* next;
+    struct Node* prev;
+};
+
+// 创建一个新的双向循环链表
+struct Node* createDoublyCircularLinkedList(int data) {
+    struct Node* newNode = (struct Node*)malloc(sizeof(struct Node));
+    newNode->data = data;
+    newNode->next = newNode;
+    newNode->prev = newNode;
+    return newNode;
+}
+
+// 在双向循环链表的末尾插入节点
+void insertAtEnd(struct Node** head, int data) {
+    struct Node* newNode = (struct Node*)malloc(sizeof(struct Node));
+    newNode->data = data;
+    newNode->next = (*head);
+    newNode->prev = (*head)->prev;
+    (*head)->prev->next = newNode;
+    (*head)->prev = newNode;
+}
+
+// 打印双向循环链表的元素
+void printDoublyCircularLinkedList(struct Node* head) {
+    if (head == NULL) {
+        printf("Empty Doubly Circular Linked List\n");
+        return;
+    }
+
+    struct Node* current = head;
+    do {
+        printf("%d ", current->data);
+        current = current->next;
+    } while (current != head);
+    printf("\n");
+}
+
+// 主函数
+int main() {
+    struct Node* head = createDoublyCircularLinkedList(1);
+
+    insertAtEnd(&head, 2);
+    insertAtEnd(&head, 3);
+
+    printf("Doubly Circular Linked List: ");c 
+    printDoublyCircularLinkedList(head);
+
+    return 0;
+}
+
+

    折半查找判定树——（快速判断某棵树是否为折半查找判定树）_折半查找树_叫我蘑菇先生的博客-CSDN博客

    对任意无序序列可建立完全二叉查找树

    先对序列排序，排序后得到升序序列为中序遍历顺序

    已知父节点i，可求出子节点下标2i和 2i+1

    -
    void make_tree(int* tree, int* a,int n,int p,int* i){
-    if(p>n||p<1||(*i)>n){
-        return;
-    }
-    make_tree(tree,a,n,p*2,i);
-    tree[p] = a[(*i)++];
-    make_tree(tree,a,n,p*2+1,i);
-}
-
    +
    void make_tree(int* tree, int* a,int n,int p,int* i){
+    if(p>n||p<1||(*i)>n){
+        return;
+    }
+    make_tree(tree,a,n,p*2,i);
+    tree[p] = a[(*i)++];
+    make_tree(tree,a,n,p*2+1,i);
+}
+

    直接找到越界为止

    diff --git "a/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK01/index.html" "b/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK01/index.html" index f3b41907..45860837 100644 --- "a/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK01/index.html" +++ "b/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK01/index.html" @@ -1490,11 +1490,11 @@

    [Example] Selection Sort : Sort a set of \(n\geq1\) integers in increasing order

    -
    for (i = 0; i < n; i++){
+
    for (i = 0; i < n; i++){
     Examine list[i] to list[n-1] and suppose that the smallest integer is at list[min];
     Interchange list[i] and list[min];
 }
-
    +

    1.1 What to Analyze

      @@ -1523,17 +1523,17 @@

      1.1 What to Analyze

    • \(T_{avg}(N)\, and\, T_{worst}(N)\) : the average and worst case time complexities as functions of input size \(N\)

    [Example] Matrix addition

    -
    void add(int a[][MAX_SIZE],
-         int b[][MAX_SIZE],
-         int c[][MAX_SIZE],
-         int rows, int cols) 
-{
-    int i, j;
-    for (i=0; i<rows; i++)/*rows+1*/
-        for (j=0;j<cols;j++)/*rows(cols+1)*/
-            c[i][j] = a[i][j]+b[i][j];/*rows*cols*/
-}
-
    +
    void add(int a[][MAX_SIZE],
+         int b[][MAX_SIZE],
+         int c[][MAX_SIZE],
+         int rows, int cols) 
+{
+    int i, j;
+    for (i=0; i<rows; i++)/*rows+1*/
+        for (j=0;j<cols;j++)/*rows(cols+1)*/
+            c[i][j] = a[i][j]+b[i][j];/*rows*cols*/
+}
+
    \[ T(rows, cols) = 2rows\times cols + 2rows+1 \]
    @@ -1541,29 +1541,29 @@

    [Example] Matrix addition

  1. 非对称

    2. [Example] Iterative function for summing a list of numbers

    -
    float sum (float list[], int n)
-{  /*add a list of numbers*/
-    float tempsum = 0; /*count = 1*/
-    int i;
-    for (i=0; i<n; i++)
-        /*count++*/
-        tempsum  += list[i]; /*count++*/
-    /*count++ for last excutaion of for*/
-   return tempsum; /*count++*/
-}
-
    +
    float sum (float list[], int n)
+{  /*add a list of numbers*/
+    float tempsum = 0; /*count = 1*/
+    int i;
+    for (i=0; i<n; i++)
+        /*count++*/
+        tempsum  += list[i]; /*count++*/
+    /*count++ for last excutaion of for*/
+   return tempsum; /*count++*/
+}
+
    \[ T_{sum}(n)=2n+3 \]

    [Example] Recursive function for summing a list of numbers

    -
    float rsum (float list[], int n)
-{/*add a list of numbers*/
-    if (n) /*count++*/
-        return rsum(list, n-1) + list[n-1];
-        /*count++*/
-    return 0; /*count++*/
-}
-
    +
    float rsum (float list[], int n)
+{/*add a list of numbers*/
+    if (n) /*count++*/
+        return rsum(list, n-1) + list[n-1];
+        /*count++*/
+    return 0; /*count++*/
+}
+
    \[ T_{rsum}(n)=2n+2 \]
    @@ -1606,17 +1606,17 @@

    [Definition]

    1-2

    [Example] Matrix addition

    -
    void add(int a[][MAX_SIZE],
-         int b[][MAX_SIZE],
-         int c[][MAX_SIZE],
-         int rows, int cols) 
-{
-    int i, j;
-    for (i=0; i<rows; i++)
-        for (j=0;j<cols;j++)
-            c[i][j] = a[i][j]+b[i][j];
-}
-
    +
    void add(int a[][MAX_SIZE],
+         int b[][MAX_SIZE],
+         int c[][MAX_SIZE],
+         int rows, int cols) 
+{
+    int i, j;
+    for (i=0; i<rows; i++)
+        for (j=0;j<cols;j++)
+            c[i][j] = a[i][j]+b[i][j];
+}
+
    \[ T(rows,cols)=\Theta(rows\cdot cols) \]
    @@ -1647,14 +1647,14 @@

    General Rules

    $$ Fib(0)=Fib(1)=1, Fib(n)=Fib(n-1)+Fib(n-2) $$

    -
    lont int Fib (int N) /*T(N)*/
-{
-    if (N<=1) /*O(1)*/
-        return 1; /*O(1)*/
-    else
-        return Fib(N-1)+Fib(N-2);
-}      /*O(1)*//*T(N-1)*//*T(N-2)*/
-
    +

    c + lont int Fib (int N) /*T(N)*/ + { + if (N<=1) /*O(1)*/ + return 1; /*O(1)*/ + else + return Fib(N-1)+Fib(N-2); + } /*O(1)*//*T(N-1)*//*T(N-2)*/

    $$ T(N)=T(N-1)+T(N-2)+2\geq Fib(N)\ \left(\frac{3}{2} \right)^n\leq Fib(N)\leq\left(\frac{5}{3}\right)^n diff --git "a/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK02/index.html" "b/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK02/index.html" index 44764870..597ace64 100644 --- "a/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK02/index.html" +++ "b/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK02/index.html" @@ -782,9 +782,9 @@

    • - + - [Definition] Data Type = {Objects} and {Operations} + [Definition] Data Type = {Objects} and @@ -1530,87 +1530,87 @@

      WEEK 2

      1.3 Compare the Algorithms

      [Example] 最大子序列和

      Algorithm 1

      -
      int  MaxSubsequenceSum ( const int A[ ],  int  N ) 
-{ 
-    int ThisSum, MaxSum, i, j, k; 
-    MaxSum = 0;   /* initialize the maximum sum */
-    for( i = 0; i < N; i++ )  /* start from A[ i ] */
-        for( j = i; j < N; j++ ) {   /* end at A[ j ] */
-            ThisSum = 0; 
-            for( k = i; k <= j; k++ ) 
-                ThisSum += A[ k ];  /* sum from A[ i ] to A[ j ] */
-            if ( ThisSum > MaxSum ) 
-                MaxSum = ThisSum;  /* update max sum */
-        }  /* end for-j and for-i */
-    return MaxSum; 
-}
-
      +
      int  MaxSubsequenceSum ( const int A[ ],  int  N ) 
+{ 
+    int ThisSum, MaxSum, i, j, k; 
+    MaxSum = 0;   /* initialize the maximum sum */
+    for( i = 0; i < N; i++ )  /* start from A[ i ] */
+        for( j = i; j < N; j++ ) {   /* end at A[ j ] */
+            ThisSum = 0; 
+            for( k = i; k <= j; k++ ) 
+                ThisSum += A[ k ];  /* sum from A[ i ] to A[ j ] */
+            if ( ThisSum > MaxSum ) 
+                MaxSum = ThisSum;  /* update max sum */
+        }  /* end for-j and for-i */
+    return MaxSum; 
+}
+
      \[ T(N)=O(N^3) \]

      Algotithm 2

      -
      int  MaxSubsequenceSum ( const int A[ ],  int  N ) 
-{ 
-    int ThisSum, MaxSum, i, j; 
-    MaxSum = 0;   /* initialize the maximum sum */
-    for( i = 0; i < N; i++ ) {   /* start from A[ i ] */
-        ThisSum = 0; 
-        for( j = i; j < N; j++ ) {   /* end at A[ j ] */
-            ThisSum += A[ j ];  /* sum from A[ i ] to A[ j ] */
-            if ( ThisSum > MaxSum ) 
-                MaxSum = ThisSum;  /* update max sum */
-        }  /* end for-j */
-    }  /* end for-i */
-    return MaxSum; 
-} 
-
      +
      int  MaxSubsequenceSum ( const int A[ ],  int  N ) 
+{ 
+    int ThisSum, MaxSum, i, j; 
+    MaxSum = 0;   /* initialize the maximum sum */
+    for( i = 0; i < N; i++ ) {   /* start from A[ i ] */
+        ThisSum = 0; 
+        for( j = i; j < N; j++ ) {   /* end at A[ j ] */
+            ThisSum += A[ j ];  /* sum from A[ i ] to A[ j ] */
+            if ( ThisSum > MaxSum ) 
+                MaxSum = ThisSum;  /* update max sum */
+        }  /* end for-j */
+    }  /* end for-i */
+    return MaxSum; 
+} 
+
      \[ T(N)=O(N^2) \]

      Algorithm 3 Divide and Conquer 分治法

      -
      static int MaxSubSum(const int A[ ], int Left, int Right)
-{
-    int MaxLeftSum, MaxRightSum;
-    int MaxLeftBorderSum, MaxRightBorderSum;
-    int LeftBorderSum, RightBorderSum;
-    int Center, i;
-
-    if (Left == Right)
-        if (A[Left] > 0)
-            return A[Left];
-        else
-            return 0;
-
-    Center = (Left + Right) / 2;
-    MaxLeftSum = MaxSubSum(A, Left, Center);
-    MaxRightSum = MaxSubSum(A, Center + 1, Right);
-
-    MaxLeftBorderSum = 0;
-    LeftBorderSum = 0;
-    for (i = Center; i >= Left; i--)
-    {
-        LeftBorderSum += A[i];
-        if (LeftBorderSum > MaxLeftBorderSum)
-            MaxLeftBorderSum = LeftBorderSum;
-    }
-
-    MaxRightBorderSum = 0;
-    RightBorderSum = 0;
-    for (i = Center+1; i <= Right; i++)
-    {
-        RightBorderSum += A[i];
-        if (RightBorderSum > MaxRightBorderSum)
-            MaxRightBorderSum = RightBorderSum;
-    }
-
-    return Max3(MaxLeftSum, MaxRightSum, MaxLeftBorderSum + MaxRightBorderSum);
-}
-
-int MaxSubsequenceSum(const int A[ ], int N)
-{
-    return MaxSubSum(A, 0, N - 1);
-}
-
      +
      static int MaxSubSum(const int A[ ], int Left, int Right)
+{
+    int MaxLeftSum, MaxRightSum;
+    int MaxLeftBorderSum, MaxRightBorderSum;
+    int LeftBorderSum, RightBorderSum;
+    int Center, i;
+
+    if (Left == Right)
+        if (A[Left] > 0)
+            return A[Left];
+        else
+            return 0;
+
+    Center = (Left + Right) / 2;
+    MaxLeftSum = MaxSubSum(A, Left, Center);
+    MaxRightSum = MaxSubSum(A, Center + 1, Right);
+
+    MaxLeftBorderSum = 0;
+    LeftBorderSum = 0;
+    for (i = Center; i >= Left; i--)
+    {
+        LeftBorderSum += A[i];
+        if (LeftBorderSum > MaxLeftBorderSum)
+            MaxLeftBorderSum = LeftBorderSum;
+    }
+
+    MaxRightBorderSum = 0;
+    RightBorderSum = 0;
+    for (i = Center+1; i <= Right; i++)
+    {
+        RightBorderSum += A[i];
+        if (RightBorderSum > MaxRightBorderSum)
+            MaxRightBorderSum = RightBorderSum;
+    }
+
+    return Max3(MaxLeftSum, MaxRightSum, MaxLeftBorderSum + MaxRightBorderSum);
+}
+
+int MaxSubsequenceSum(const int A[ ], int N)
+{
+    return MaxSubSum(A, 0, N - 1);
+}
+

      $$ \because T(N)=2T(\frac N2)+cN\quad T(1)=O(1)\ T(\frac N2)=2T(\frac N {2^2})+c\frac N2\ @@ -1619,20 +1619,20 @@

      [Example] 最大子序列和

      \therefore T(N)=2^kT(\frac N{2^k})+kcN=N\cdot O(1)+cN\log N $$ Algorithm 4 On-line Algorithm 在线算法

      -
      int MaxSubsequenceSum( const int  A[ ],  int  N ) 
-{ 
-    int ThisSum, MaxSum, j; 
-    ThisSum = MaxSum = 0; 
-    for ( j = 0; j < N; j++ ) { 
-        ThisSum += A[ j ]; 
-        if ( ThisSum > MaxSum ) 
-            MaxSum = ThisSum; 
-        else if ( ThisSum < 0 ) 
-            ThisSum = 0;
-    }  /* end for-j */
-    return MaxSum; 
-} 
-
      +
      int MaxSubsequenceSum( const int  A[ ],  int  N ) 
+{ 
+    int ThisSum, MaxSum, j; 
+    ThisSum = MaxSum = 0; 
+    for ( j = 0; j < N; j++ ) { 
+        ThisSum += A[ j ]; 
+        if ( ThisSum > MaxSum ) 
+            MaxSum = ThisSum; 
+        else if ( ThisSum < 0 ) 
+            ThisSum = 0;
+    }  /* end for-j */
+    return MaxSum; 
+} 
+
      \[ T(N)=O(N) \]
      @@ -1646,49 +1646,49 @@

      1.4 Logrithms in the Running Time

    • 如果一个算法用常数时间将问题的大小削减为其一部分(通常是1/2)，那么该算法就是\(O(logN)\)的
    -
    int BinarySearch ( const ElementType A[ ], ElementType X, int N ) 
-{ 
-    int  Low, Mid, High; 
-    Low = 0;  High = N - 1; 
-    while ( Low <= High ) { 
-        Mid = ( Low + High ) / 2; 
-        if ( A[ Mid ] < X ) 
-            Low = Mid + 1; 
-        else 
-            if ( A[ Mid ] > X ) 
-                High = Mid - 1; 
-            else 
-                return  Mid; /* Found */ 
-    }  /* end while */
-    return  NotFound; /* NotFound is defined as -1 */ 
-} 
-
    +
    int BinarySearch ( const ElementType A[ ], ElementType X, int N ) 
+{ 
+    int  Low, Mid, High; 
+    Low = 0;  High = N - 1; 
+    while ( Low <= High ) { 
+        Mid = ( Low + High ) / 2; 
+        if ( A[ Mid ] < X ) 
+            Low = Mid + 1; 
+        else 
+            if ( A[ Mid ] > X ) 
+                High = Mid - 1; 
+            else 
+                return  Mid; /* Found */ 
+    }  /* end while */
+    return  NotFound; /* NotFound is defined as -1 */ 
+} 
+
    \[ T_{worst}(N)=O(\log N) \]

    [Example] Euclid’s Algorithm

    -
    int Gcd(int M, int N)
-{
-    int Rem;
-
-    while (N > 0)
-    {
-        Rem = M % N;
-        M = N;
-        N = Rem;
-    }
-    return M;
-}
-
    +
    int Gcd(int M, int N)
+{
+    int Rem;
+
+    while (N > 0)
+    {
+        Rem = M % N;
+        M = N;
+        N = Rem;
+    }
+    return M;
+}
+

    [Example] Efficient exponentiation

    -
    long int Pow(long int X, int N)
-{
-    if (N == 0) return 1;
-    if (N == 1) return X;
-    if (IsEven(N)) return Pow(X*X, N/2);/*return Pow(X, N/2)*Pow(X, N/2) affects the efficiency*/
-    else return Pow(X*X, N/2)*X; /*return Pow(X, N-1)*X is the same*/
-}
-
    +
    long int Pow(long int X, int N)
+{
+    if (N == 0) return 1;
+    if (N == 1) return X;
+    if (IsEven(N)) return Pow(X*X, N/2);/*return Pow(X, N/2)*Pow(X, N/2) affects the efficiency*/
+    else return Pow(X*X, N/2)*X; /*return Pow(X, N-1)*X is the same*/
+}
+

    1.5 Checking Your Analysis

    Method 1

    @@ -1700,7 +1700,7 @@

    Method 2

    2 LIst, Stacks and Queues

    2.1 Abstract Data Type(ADT) 抽象数据类型

    -

    [Definition] Data Type = {Objects} and {Operations}

    +

    [Definition] Data Type = {Objects} and

    [Definition] An Abstract Data Type(ADT) is a data type that is organized in such a way that the specification on the objects and specification of the operations on the objects are separated from the representation of the objects and the implementation on the operations.

    2.2 The List ADT

    @@ -1751,87 +1751,92 @@

    Linked Lists

    Finding take \(O(N)\) times.

    -
    /*Return true if L is empty*/
-int IsEmpty(List L)
-{
-  return L->Next == NULL;
-}
-
    -
    /*Return true if P is the last position in list L*/
-/*Parameter L is unused in this implementation*/
-int IsLast(Position P, List L)
-{
-  return P->Next == NULL;
-}
-
    -
    /*Return Position of X in L; NULL if not found*/
-Position Find(Element X, List L)
-{
-  Position P;
-
-  P = L->Next;
-  while (P != NULL && P->Element != X) P = P->Next;
-
-  return P;
-}
-
    -
    /*Delete first occurence of X from a list*/
-/*Assume use of a header node*/
-void Delete(ElementType X, List L)
-{
-  Position P, TmpCell;
-
-  P = FindPrevious(X, L);
-
-  if (!IsLast(P, L))
-  {
-      TmpCell = P->Next;
-      P->Next = TmpCell->Next;
-      free(TmpCell);
-  }
-}
-
    -
    /*If X is not found, then Next field of returned*/
-/*Assumes a header*/
-Position FindPrevious(ElementType X, List L)
-{
-  Position P;
-
-  P = L;
-  while (P->Next != NULL && P->Next->Element != X) P = P->Next;
-
-  return P;
-}
-
    -
    /*Insert (after legal position P)*/
-/*Header implementation assumed*/
-/*Parameter L is unused in this implementation*/
-void Insert(ElementType X, List L, Position P)
-{
-  Position TmpCell;
-
-  TmpCell = malloc(sizeof(struct Node));
-  if (TmpCell == NULL) FatalError("Out of space!")
-
-  TmpCell->Element = X;
-  TmpeCell->Next = P->Next;
-  P->Next = TmpCell;
-}
-
    -
    void DeleteList(List L)
-{
-  Position P, Tmp;
-
-  P = L->Next;
-  L->Next = NULL;
-  while (P != NULL)
-  {
-      Tmp = P->Next;
-      free(P);
-      P = Tmp;
-  }
-}
-
    +

    c + /*Return true if L is empty*/ + int IsEmpty(List L) + { + return L->Next == NULL; + }

    +

    c + /*Return true if P is the last position in list L*/ + /*Parameter L is unused in this implementation*/ + int IsLast(Position P, List L) + { + return P->Next == NULL; + }

    +

    ```c + /Return Position of X in L; NULL if not found/ + Position Find(Element X, List L) + { + Position P;

    +
    P = L->Next;
+while (P != NULL && P->Element != X) P = P->Next;
+
+return P;
+
    +

    } + ```

    +

    ```c + /Delete first occurence of X from a list/ + /Assume use of a header node/ + void Delete(ElementType X, List L) + { + Position P, TmpCell;

    +
    P = FindPrevious(X, L);
+
+if (!IsLast(P, L))
+{
+    TmpCell = P->Next;
+    P->Next = TmpCell->Next;
+    free(TmpCell);
+}
+
    +

    } + ```

    +

    ```c + /If X is not found, then Next field of returned/ + /Assumes a header/ + Position FindPrevious(ElementType X, List L) + { + Position P;

    +
    P = L;
+while (P->Next != NULL && P->Next->Element != X) P = P->Next;
+
+return P;
+
    +

    } + ```

    +

    ```c + /Insert (after legal position P)/ + /Header implementation assumed/ + /Parameter L is unused in this implementation/ + void Insert(ElementType X, List L, Position P) + { + Position TmpCell;

    +
    TmpCell = malloc(sizeof(struct Node));
+if (TmpCell == NULL) FatalError("Out of space!")
+
+TmpCell->Element = X;
+TmpeCell->Next = P->Next;
+P->Next = TmpCell;
+
    +

    } + ```

    +

    ```c + void DeleteList(List L) + { + Position P, Tmp;

    +
    P = L->Next;
+L->Next = NULL;
+while (P != NULL)
+{
+    Tmp = P->Next;
+    free(P);
+    P = Tmp;
+}
+
    +

    } + ```

    Doubly Linked Circular Lists

    • Finding take \(O(\frac N 2)\) times.
      • @@ -1862,58 +1867,60 @@

      Two Applications

      [Representation 1]

    二分法的细节加细节 你真的应该搞懂！！！_二分算法-CSDN博客

-
typedef struct {
-  int CoeffArray [ MaxDegree + 1 ] ;
-  int HighPower;
-}  *Polynomial ; 
-
-
/*将多项式初始化为零*/
-void ZeroPolynomial(Polynomial Poly)
-{
-  int i;
-  for(i = O; i <= MaxDegree; i++)
-      Poly->CoeffArray[ i ] = O;
-  Poly->HighPower = O;
-}
-
-
/*两个多项式相加*/
-void AddPolynomial(const Polynomial Poly1, const Polynomial Poly2, Polynomial PolySum)
-{
-    int i;
-
-    ZeroPolynomial(PolySum);
-  PolySum->HighPower = Max(Poly1->HighPower, Poly2->HighPower);
-
-    for (i = PolySum->HighPower; i >= O; i--)
-      PolySum->CoeffArray[ i ] = Poly1->CoeffArray[ i ] + Poly2->CoeffArray[ i ];
-}
-
-
void MultPolynomial(const Polynomial Poly1, const Polynomial Poly2, Polynomial PolyProd)
-{
-    int i, j;
-
-    ZeroPolynomial (PolyProd);
-  PolyProd->HighPower = Poly1->HighPower + Poly2->HighPower;
-
-    if(PolyProd->HighPower > MaxDegree)
-      Error("Exceeded array size");
-  else
-      for(i = O; i <= Poly1->HighPower; i++)
-          for(j = O; j <= Poly2->HighPower; j++)
-              PolyProd->CoeffArray[ i + j ] += Poly1->CoeffArray[ i ] * Poly2->CoeffArray[ j ];
-}
-
+

c + typedef struct { + int CoeffArray [ MaxDegree + 1 ] ; + int HighPower; + } *Polynomial ;

+

c + /*将多项式初始化为零*/ + void ZeroPolynomial(Polynomial Poly) + { + int i; + for(i = O; i <= MaxDegree; i++) + Poly->CoeffArray[ i ] = O; + Poly->HighPower = O; + }

+

```c + /两个多项式相加/ + void AddPolynomial(const Polynomial Poly1, const Polynomial Poly2, Polynomial PolySum) + { + int i;

+
  ZeroPolynomial(PolySum);
+PolySum->HighPower = Max(Poly1->HighPower, Poly2->HighPower);
+
+  for (i = PolySum->HighPower; i >= O; i--)
+    PolySum->CoeffArray[ i ] = Poly1->CoeffArray[ i ] + Poly2->CoeffArray[ i ];
+
+

} + ```

+

```c + void MultPolynomial(const Polynomial Poly1, const Polynomial Poly2, Polynomial PolyProd) + { + int i, j;

+
  ZeroPolynomial (PolyProd);
+PolyProd->HighPower = Poly1->HighPower + Poly2->HighPower;
+
+  if(PolyProd->HighPower > MaxDegree)
+    Error("Exceeded array size");
+else
+    for(i = O; i <= Poly1->HighPower; i++)
+        for(j = O; j <= Poly2->HighPower; j++)
+            PolyProd->CoeffArray[ i + j ] += Poly1->CoeffArray[ i ] * Poly2->CoeffArray[ j ];
+
+

} + ```

  • [Representation 2]
-
typedef struct poly_node *poly_ptr;
-struct poly_node{
-    int Coefficient;  /* assume coefficients are integers */
-    int Exponent;
-    poly_ptr Next;
-};
-typedef poly_ptr a;    /* nodes sorted by exponent */
-
+

c + typedef struct poly_node *poly_ptr; + struct poly_node{ + int Coefficient; /* assume coefficients are integers */ + int Exponent; + poly_ptr Next; + }; + typedef poly_ptr a; /* nodes sorted by exponent */

  • 只存储非零项

    diff --git "a/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK03/index.html" "b/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK03/index.html" index a594d532..646796b4 100644 --- "a/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK03/index.html" +++ "b/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK03/index.html" @@ -1406,73 +1406,73 @@

    Linked List Implementatio
    • The calls to malloc and free are expensive. Simply keep another stack as a recycle bin.
    -
    int IsEmpty(Stack S)
-{
-  return S->Next == NULL;
-}
-
    -
    Stack CreateStack(void)
-{
-  Stack S;
-  S = malloc(sizeof(struct Node));
-  if (S == NULL)
-      Fatal Error("Out of space!");
-  S->Next == NULL;
-  MakeEmpty(S);
-  return S;
-}
-
-void MakeEmpty(Stack S)
-{
-  if (S == NULL)
-      Error("Must use CreateStack first");
-  else
-      while(!IsEmpty(S)) Pop(S);
-}
-
    -
    void Push(ElementType X, Stack S)
-{
-  PtrToNode TmpCell;
-  TmpCell = malloc(sizeof(struct Node));
-  if (TmpCell == NULL)
-      Fatal Error("Out of space!") ;
-  else
-  {
-      TmpCell->Element = X;
-      TmpCe11->Next = S->Next;
-      S->Next = TmpCell;
-  }
-}
-
    -
    ElementType Top(Stack S)
-{
-  if(!IsEmpty(S))
-      return S->Next->Element;
-  Error("Empty stack") ;
-  return O; /* Return value used to avoid warning*/
-}
-
    -
    void Pop(Stack s)
-{
-  PtrToNode FirstCell;
-  if(IsEmpty(S))
-      Error("Empty stack") ;
-  else
-  {
-      FirstCe11 = S->Next;
-      S->Next = S->Next->Next;
-      free(FirstCe11);
-  }
-}
-
    +

    c + int IsEmpty(Stack S) + { + return S->Next == NULL; + }

    +

    ```c + Stack CreateStack(void) + { + Stack S; + S = malloc(sizeof(struct Node)); + if (S == NULL) + Fatal Error("Out of space!"); + S->Next == NULL; + MakeEmpty(S); + return S; + }

    +

    void MakeEmpty(Stack S) + { + if (S == NULL) + Error("Must use CreateStack first"); + else + while(!IsEmpty(S)) Pop(S); + } + ```

    +

    c + void Push(ElementType X, Stack S) + { + PtrToNode TmpCell; + TmpCell = malloc(sizeof(struct Node)); + if (TmpCell == NULL) + Fatal Error("Out of space!") ; + else + { + TmpCell->Element = X; + TmpCe11->Next = S->Next; + S->Next = TmpCell; + } + }

    +

    c + ElementType Top(Stack S) + { + if(!IsEmpty(S)) + return S->Next->Element; + Error("Empty stack") ; + return O; /* Return value used to avoid warning*/ + }

    +

    c + void Pop(Stack s) + { + PtrToNode FirstCell; + if(IsEmpty(S)) + Error("Empty stack") ; + else + { + FirstCe11 = S->Next; + S->Next = S->Next->Next; + free(FirstCe11); + } + }

    Array Implementation of Stacks

    -
    struct StackRecord {
-    int Capacity;          /* size of stack */
-    int TopOfStack;        /* the top pointer */
-    /* ++ for push, -- for pop, -1 for empty stack */
-    ElementType *Array;    /* array for stack elements */
-}; 
-
    +
    struct StackRecord {
+    int Capacity;          /* size of stack */
+    int TopOfStack;        /* the top pointer */
+    /* ++ for push, -- for pop, -1 for empty stack */
+    ElementType *Array;    /* array for stack elements */
+}; 
+

    • The stack model must be well encapsulated(封装). That is, no part of your code, except for the stack routines, can attempt to access the Array or TopOfStack variable.

      @@ -1481,95 +1481,96 @@

      Array Implementation of Stacks

      Error check must be done before Push or Pop (Top).

    -
    Stack CreateStack(int MaxElements)
-{
-  Stack S;
-  if(MaxElements < MinStackSize)
-  Error("Stack size is too small") ;
-  S = malloc(sizeof(struct StackRecord));
-  if (S == NULL)
-      Fatal Error("Out of space!!!") ;
-
-  S->Array = malloc(sizeof(ElementType) * MaxElements) ;
-  if(S->Array = NULL)
-      Fatal Error("Out of space!!!");
-  S->Capacity = MaxElements;
-  MakeEmpty(S) ;
-  return S;
-}
-
    -
    void DisposeStack(Stack S)
-{
-  if(S != NULL)
-  {
-      free(S->Array);
-      free(S);
-  }
-}
-
    -
    int IsEmpty(Stack S)
-{
-  return S->TopOfStack == EmptyTOS;
-}
-
    -
    void MakeEmpty(Stack S)
-{
-  S->TopOfStack = EmptyTOS;
-}
-
    -
    void Push(ElementType X, Stack S)
-{
-  if (IsFull(S))
-      Error("Full stack");
-  else
-      S->Array[ ++S->TopOfStack ] = X;
-}
-
    -
    ElementType Top(Stack S)
-{
-  if(! IsEmpty(S))
-      return S->Array[ S->TopOfStack ];
-  Error("Empty stack") ;
-  return O; /* Return value used to avoid warning*/
-}
-
    -
    void Pop(Stack S)
-{
-  if(IsEmpty(S))
-      Error("Empty stack") ;
-  else
-      S->TopOfStack--;
-}
-
    -
    ElementType TopAndPop(Stack S)
-{
-  if(!Is Empty(S))
-      return S->Array[ S->TopOfStack-- ];
-  Error("Empty stack");
-  return O; /* Return value used to avoid warnin */
-}
-
    +

    ```c + Stack CreateStack(int MaxElements) + { + Stack S; + if(MaxElements < MinStackSize) + Error("Stack size is too small") ; + S = malloc(sizeof(struct StackRecord)); + if (S == NULL) + Fatal Error("Out of space!!!") ;

    +
    S->Array = malloc(sizeof(ElementType) * MaxElements) ;
+if(S->Array = NULL)
+    Fatal Error("Out of space!!!");
+S->Capacity = MaxElements;
+MakeEmpty(S) ;
+return S;
+
    +

    } + ```

    +

    c + void DisposeStack(Stack S) + { + if(S != NULL) + { + free(S->Array); + free(S); + } + }

    +

    c + int IsEmpty(Stack S) + { + return S->TopOfStack == EmptyTOS; + }

    +

    c + void MakeEmpty(Stack S) + { + S->TopOfStack = EmptyTOS; + }

    +

    c + void Push(ElementType X, Stack S) + { + if (IsFull(S)) + Error("Full stack"); + else + S->Array[ ++S->TopOfStack ] = X; + }

    +

    c + ElementType Top(Stack S) + { + if(! IsEmpty(S)) + return S->Array[ S->TopOfStack ]; + Error("Empty stack") ; + return O; /* Return value used to avoid warning*/ + }

    +

    c + void Pop(Stack S) + { + if(IsEmpty(S)) + Error("Empty stack") ; + else + S->TopOfStack--; + }

    +

    c + ElementType TopAndPop(Stack S) + { + if(!Is Empty(S)) + return S->Array[ S->TopOfStack-- ]; + Error("Empty stack"); + return O; /* Return value used to avoid warnin */ + }

    Application

    1. Balancing Symbols

    检查括号是否平衡

    -
    Algorithm  {
-    Make an empty stack S;
-    while (read in a character c) {
-        if (c is an opening symbol)
-            Push(c, S);
-        else if (c is a closing symbol) {
-            if (S is empty)  { ERROR; exit; }
-            else  {  /* stack is okay */
-                if  (Top(S) doesn’t match c)  { ERROR, exit; }
-                else  Pop(S);
-            }  /* end else-stack is okay */
-        }  /* end else-if-closing symbol */
-    } /* end while-loop */ 
-    if (S is not empty)  ERROR;
-}
-
    +

    pseudocode + Algorithm { + Make an empty stack S; + while (read in a character c) { + if (c is an opening symbol) + Push(c, S); + else if (c is a closing symbol) { + if (S is empty) { ERROR; exit; } + else { /* stack is okay */ + if (Top(S) doesn’t match c) { ERROR, exit; } + else Pop(S); + } /* end else-stack is okay */ + } /* end else-if-closing symbol */ + } /* end while-loop */ + if (S is not empty) ERROR; + }

    1. Postfix Evaluation 后缀表达式

      @@ -1611,14 +1612,14 @@

      2.4 The Queue ADT

    2. Dequeue

Array Implementation of Queues

-
struct QueueRecord {
-    int Capacity ;       /* max size of queue */
-    int Front;           /* the front pointer */
-    int Rear;            /* the rear pointer */
-    int Size;            /* Optional - the current size of queue */
-    ElementType *Array;  /* array for queue elements */
- }; 
-
+
struct QueueRecord {
+    int Capacity ;       /* max size of queue */
+    int Front;           /* the front pointer */
+    int Rear;            /* the rear pointer */
+    int Size;            /* Optional - the current size of queue */
+    ElementType *Array;  /* array for queue elements */
+ }; 
+

Circular Queue :

3-33-4

    diff --git "a/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK04/index.html" "b/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK04/index.html" index 25e7e048..5eedd1bd 100644 --- "a/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK04/index.html" +++ "b/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK04/index.html" @@ -1479,73 +1479,73 @@

    Tree Traversals (visit eac
    1. Preorder Traversal 先处理根节点，再处理左右节点，因此preorder中第一个遍历到的为根节点
    -
    void preorder( tree_ptr tree )
-{ 
-    if( tree )   
-    {
-        visit ( tree );
-        for (each child C of tree )
-            preorder ( C );
-    }
-}
-
    +

    pseudocode + void preorder( tree_ptr tree ) + { + if( tree ) + { + visit ( tree ); + for (each child C of tree ) + preorder ( C ); + } + }

    1. Postorder Traversal先处理左右子树，根节点最后返回，因此postorder中根节点最后返回
    -
    void postorder( tree_ptr tree )
-{  
- if( tree )   
- {
-        for (each child C of tree )
-         postorder ( C );
-        visit ( tree );
-    }
-}
-
    +

    pseudocode + void postorder( tree_ptr tree ) + { + if( tree ) + { + for (each child C of tree ) + postorder ( C ); + visit ( tree ); + } + }

    1. Levelorder Traversal 层序遍历，使用一个queue
    -
    void levelorder( tree_ptr tree )
-{   
- enqueue ( tree );
-    while (queue is not empty) 
-    {
-        visit ( T = dequeue ( ) );
-        for (each child C of T )
-            enqueue ( C );
-    }
-}
-
    +

    pseudocode + void levelorder( tree_ptr tree ) + { + enqueue ( tree ); + while (queue is not empty) + { + visit ( T = dequeue ( ) ); + for (each child C of T ) + enqueue ( C ); + } + }

    1. Inorder Traversal 返回顺序为左、根、右，中序遍历最为关键
    -
    void inorder( tree_ptr  tree )
-{  
-     if( tree )   
- {
-     inorder ( tree->Left );
-         visit ( tree->Element );
-         inorder ( tree->Right );
-   }
-}
-
    +

    pseudocode + void inorder( tree_ptr tree ) + { + if( tree ) + { + inorder ( tree->Left ); + visit ( tree->Element ); + inorder ( tree->Right ); + } + }

    Iterative Program : 使用自己定义的栈替换系统的栈，实现中序遍历

    leftmost 的左节点和rightmost的右节点都是NULL

    -
    void iter_inorder( tree_ptr tree )
-{ 
- Stack  S = CreateStack( MAX_SIZE );
- for ( ; ; )  
- {
-     for ( ; tree; tree = tree->Left )
-         Push ( tree, S );
-         tree = Top ( S );  
-         Pop( S );
-         if ( !tree ) break;
-         visit ( tree->Element );
-         tree = tree->Right; 
-    }
-}
-
    +

    pseudocode + void iter_inorder( tree_ptr tree ) + { + Stack S = CreateStack( MAX_SIZE ); + for ( ; ; ) + { + for ( ; tree; tree = tree->Left ) + Push ( tree, S ); + tree = Top ( S ); + Pop( S ); + if ( !tree ) break; + visit ( tree->Element ); + tree = tree->Right; + } + }

    4-3

    Threaded Binary Trees

      @@ -1566,17 +1566,17 @@

      Threaded Binary Trees

    • There must not be any loose threads. Therefore a threaded binary tree must have a head node of which the left child points to the first node.（右节点指针指向自身）
    -
    typedef struct ThreadedTreeNode *PtrToThreadedNode;
-typedef struct PtrToThreadedNode ThreadedTree;
-typedef struct ThreadedTreeNode 
-{
-    int LeftThread;        /* if it is TRUE, then Left */
-    ThreadedTree Left;     /* is a thread, not a child ptr.*/
-    ElementType Element;
-    int RightThread;       /* if it is TRUE, then Right */
-    ThreadedTree Right;    /* is a thread, not a child ptr.*/
-}
-
    +
    typedef struct ThreadedTreeNode *PtrToThreadedNode;
+typedef struct PtrToThreadedNode ThreadedTree;
+typedef struct ThreadedTreeNode 
+{
+    int LeftThread;        /* if it is TRUE, then Left */
+    ThreadedTree Left;     /* is a thread, not a child ptr.*/
+    ElementType Element;
+    int RightThread;       /* if it is TRUE, then Right */
+    ThreadedTree Right;    /* is a thread, not a child ptr.*/
+}
+
    • 线索化的实质就是将二叉链表中的空指针改为指向前驱或后继的线索。由于前驱和后继信息只有在遍历该二叉树时才能得到，所以，线索化的过程就是在遍历的过程中修改空指针的过程。
    diff --git "a/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK05/index.html" "b/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK05/index.html" index 7c8a765b..cf280e61 100644 --- "a/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK05/index.html" +++ "b/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK05/index.html" @@ -1410,86 +1410,86 @@

    Implementations

    1. Find
    -
    Position Find( ElementType X, SearchTree T ) 
-{ 
- if ( T == NULL ) 
-     return NULL;  /* not found in an empty tree */
-    if ( X < T->Element )  /* if smaller than root */
-        return Find( X, T->Left );  /* search left subtree */
-    else 
-     if ( X > T->Element )  /* if larger than root */
-         return  Find( X, T->Right );  /* search right subtree */
-        else   /* if X == root */
-         return  T;  /* found */
-} 
-
    +

    pseudocode + Position Find( ElementType X, SearchTree T ) + { + if ( T == NULL ) + return NULL; /* not found in an empty tree */ + if ( X < T->Element ) /* if smaller than root */ + return Find( X, T->Left ); /* search left subtree */ + else + if ( X > T->Element ) /* if larger than root */ + return Find( X, T->Right ); /* search right subtree */ + else /* if X == root */ + return T; /* found */ + }

    • \(T(N)=S(N)=O(d)\) where \(d\) is the depth of X

    Iterative program :

    -
    Position Iter_Find( ElementType X, SearchTree T ) 
-{ 
- while ( T )
- {
-     if ( X == T->Element )  
-         return T;  /* found */
-        if ( X < T->Element )
-            T = T->Left; /*move down along left path */
-        else
-             T = T-> Right; /* move down along right path */
-    }  /* end while-loop */
-    return NULL;   /* not found */
-} 
-
    +

    pseudocode + Position Iter_Find( ElementType X, SearchTree T ) + { + while ( T ) + { + if ( X == T->Element ) + return T; /* found */ + if ( X < T->Element ) + T = T->Left; /*move down along left path */ + else + T = T-> Right; /* move down along right path */ + } /* end while-loop */ + return NULL; /* not found */ + }

    1. FindMin
    -
    Position FindMin( SearchTree T ) 
-{ 
- if ( T == NULL )   
-     return NULL; /* not found in an empty tree */
-    else 
-        if ( T->Left == NULL ) return T;  /* found left most */
-        else return FindMin( T->Left );   /* keep moving to left */
-} 
-
    +

    pseudocode + Position FindMin( SearchTree T ) + { + if ( T == NULL ) + return NULL; /* not found in an empty tree */ + else + if ( T->Left == NULL ) return T; /* found left most */ + else return FindMin( T->Left ); /* keep moving to left */ + }

    1. FindMax
    -
    Position FindMax( SearchTree T ) 
-{ 
- if ( T != NULL ) 
-     while ( T->Right != NULL )   
-         T = T->Right;   /* keep moving to find right most */
-    return T;  /* return NULL or the right most */
-} 
-
    +

    pseudocode + Position FindMax( SearchTree T ) + { + if ( T != NULL ) + while ( T->Right != NULL ) + T = T->Right; /* keep moving to find right most */ + return T; /* return NULL or the right most */ + }

    1. Insert
    -
    SearchTree Insert( ElementType X, SearchTree T ) 
-{ 
-    if ( T == NULL ) /* Create and return a one-node tree */ 
-    { 
-     T = malloc( sizeof( struct TreeNode ) ); 
-     if ( T == NULL ) 
-         FatalError( "Out of space!!!" ); 
-     else 
-     { 
-         T->Element = X; 
-         T->Left = T->Right = NULL; 
-     } 
-    }  /* End creating a one-node tree */
-    else  /* If there is a tree */
-         if ( X < T->Element ) 
-         T->Left = Insert( X, T->Left ); 
-     else 
-         if ( X > T->Element ) 
-             T->Right = Insert( X, T->Right ); 
-     /* Else X is in the tree already; we'll do nothing */ 
-    return  T;   /* Do not forget this line!! */ 
-}
-
    +

    pseudocode + SearchTree Insert( ElementType X, SearchTree T ) + { + if ( T == NULL ) /* Create and return a one-node tree */ + { + T = malloc( sizeof( struct TreeNode ) ); + if ( T == NULL ) + FatalError( "Out of space!!!" ); + else + { + T->Element = X; + T->Left = T->Right = NULL; + } + } /* End creating a one-node tree */ + else /* If there is a tree */ + if ( X < T->Element ) + T->Left = Insert( X, T->Left ); + else + if ( X > T->Element ) + T->Right = Insert( X, T->Right ); + /* Else X is in the tree already; we'll do nothing */ + return T; /* Do not forget this line!! */ + }

    • 内存越界后不会马上报错，在下一次free或malloc时会失败
    • Handle duplicated keys
      • @@ -1505,32 +1505,32 @@

      Implementations

    • Delete a degree 1 node : Replace the node by its single child
    • Delete a degree 2 node : 用左子树最大值结点或右子树最小值结点替换
    -
    SearchTree Delete( ElementType X, SearchTree T ) 
-{    
- Position TmpCell; 
-    if ( T == NULL ) Error( "Element not found" ); 
-    else if ( X < T->Element )  /* Go left */ 
-     T->Left = Delete( X, T->Left ); 
-    else if ( X > T->Element )  /* Go right */ 
-     T->Right = Delete( X, T->Right ); 
- else  /* Found element to be deleted */ 
-     if ( T->Left && T->Right ) {  /* Two children */ 
-     /* Replace with smallest in right subtree */ 
-         TmpCell = FindMin( T->Right ); 
-         T->Element = TmpCell->Element; 
-         T->Right = Delete( T->Element, T->Right );  } /* End if */
-     else 
-     {  /* One or zero child */ 
-         TmpCell = T; 
-         if ( T->Left == NULL ) /* Also handles 0 child */ 
-             T = T->Right; 
-         else if ( T->Right == NULL )  
-             T = T->Left; 
-         free( TmpCell );  
-      }  /* End else 1 or 0 child */
-      return  T; 
-}
-
    +

    pseudocode + SearchTree Delete( ElementType X, SearchTree T ) + { + Position TmpCell; + if ( T == NULL ) Error( "Element not found" ); + else if ( X < T->Element ) /* Go left */ + T->Left = Delete( X, T->Left ); + else if ( X > T->Element ) /* Go right */ + T->Right = Delete( X, T->Right ); + else /* Found element to be deleted */ + if ( T->Left && T->Right ) { /* Two children */ + /* Replace with smallest in right subtree */ + TmpCell = FindMin( T->Right ); + T->Element = TmpCell->Element; + T->Right = Delete( T->Element, T->Right ); } /* End if */ + else + { /* One or zero child */ + TmpCell = T; + if ( T->Left == NULL ) /* Also handles 0 child */ + T = T->Right; + else if ( T->Right == NULL ) + T = T->Left; + free( TmpCell ); + } /* End else 1 or 0 child */ + return T; + }

    • \(T(N)=O(d)\)
    diff --git "a/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK06/index.html" "b/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK06/index.html" index 6142643f..ddb04065 100644 --- "a/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK06/index.html" +++ "b/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK06/index.html" @@ -1589,24 +1589,24 @@

    Structure Property

    None && {2i+1>n}\\ \end{array} \right.\) -
    PriorityQueue Initialize( int MaxElements ) 
-{ 
-    PriorityQueue H; 
-    if ( MaxElements < MinPQSize ) 
-        return Error( "Priority queue size is too small" ); 
-    H = malloc(sizeof( struct HeapStruct )); 
-    if ( H == NULL ) 
-        return FatalError( "Out of space!!!" ); 
-    /* Allocate the array plus one extra for sentinel */ 
-    H->Elements = malloc(( MaxElements + 1 ) * sizeof( ElementType )); 
-    if ( H->Elements == NULL ) 
-        return FatalError( "Out of space!!!" ); 
-    H->Capacity = MaxElements; 
-    H->Size = 0; 
-    H->Elements[0] = MinData;  /* set the sentinel */
-    return H; 
-}
-
    +
    PriorityQueue Initialize( int MaxElements ) 
+{ 
+    PriorityQueue H; 
+    if ( MaxElements < MinPQSize ) 
+        return Error( "Priority queue size is too small" ); 
+    H = malloc(sizeof( struct HeapStruct )); 
+    if ( H == NULL ) 
+        return FatalError( "Out of space!!!" ); 
+    /* Allocate the array plus one extra for sentinel */ 
+    H->Elements = malloc(( MaxElements + 1 ) * sizeof( ElementType )); 
+    if ( H->Elements == NULL ) 
+        return FatalError( "Out of space!!!" ); 
+    H->Capacity = MaxElements; 
+    H->Size = 0; 
+    H->Elements[0] = MinData;  /* set the sentinel */
+    return H; 
+}
+

    Heap Order Property

    [Definition] A min tree is a tree in which the key value in each node is no larger than the key values in its children (if any). A min heap is a complete binary tree that is also a min tree.

      @@ -1616,51 +1616,51 @@

      Basic Heap Operations

      1. Insertion
      -
      /*H->Element[ 0 ] is a sentinel that is no larger than the minimum element in the heap.*/ 
-void Insert( ElementType X, PriorityQueue H ) 
-{ 
- int i; 
-    if ( IsFull( H )) 
-    { 
-     Error( "Priority queue is full" ); 
-     return; 
-    } 
-    for ( i = ++H->Size; H->Elements[ i/2 ] > X; i /= 2 ) 
-     H->Elements[ i ] = H->Elements[ i/2 ]; /*Percolate up, faster than swap*/
-    H->Elements[ i ] = X; 
-}
-
      +

      c + /*H->Element[ 0 ] is a sentinel that is no larger than the minimum element in the heap.*/ + void Insert( ElementType X, PriorityQueue H ) + { + int i; + if ( IsFull( H )) + { + Error( "Priority queue is full" ); + return; + } + for ( i = ++H->Size; H->Elements[ i/2 ] > X; i /= 2 ) + H->Elements[ i ] = H->Elements[ i/2 ]; /*Percolate up, faster than swap*/ + H->Elements[ i ] = X; + }

      $$ T(N)=O(\log N) $$

      1. DeleteMin
      -

      ElementType DeleteMin( PriorityQueue H ) 
-{ 
-    int i, Child; 
-    ElementType MinElement, LastElement; 
-    if ( IsEmpty( H ) ) 
-    { 
-        Error( "Priority queue is empty" ); 
-        return H->Elements[ 0 ];   
-    } 
-    MinElement = H->Elements[ 1 ];  /*Save the min element*/
-    LastElement = H->Elements[ H->Size-- ];  /*Take last and reset size*/
-    for ( i = 1; i * 2 <= H->Size; i = Child )  /*Find smaller child*/ 
-    {
-        Child = i * 2; 
-        if (Child != H->Size && H->Elements[Child+1] < H->Elements[Child]) 
-             Child++;     
-        if ( LastElement > H->Elements[ Child ] )   /*Percolate one level*/ 
-             H->Elements[ i ] = H->Elements[ Child ]; 
-        else     
-         break;   /*Find the proper position*/
-    } 
-    H->Elements[ i ] = LastElement; 
-    return MinElement; 
-}
-
      +

      c + ElementType DeleteMin( PriorityQueue H ) + { + int i, Child; + ElementType MinElement, LastElement; + if ( IsEmpty( H ) ) + { + Error( "Priority queue is empty" ); + return H->Elements[ 0 ]; + } + MinElement = H->Elements[ 1 ]; /*Save the min element*/ + LastElement = H->Elements[ H->Size-- ]; /*Take last and reset size*/ + for ( i = 1; i * 2 <= H->Size; i = Child ) /*Find smaller child*/ + { + Child = i * 2; + if (Child != H->Size && H->Elements[Child+1] < H->Elements[Child]) + Child++; + if ( LastElement > H->Elements[ Child ] ) /*Percolate one level*/ + H->Elements[ i ] = H->Elements[ Child ]; + else + break; /*Find the proper position*/ + } + H->Elements[ i ] = LastElement; + return MinElement; + } $$ T(N)=O(\log N) $$

      @@ -1683,9 +1683,9 @@

      Other Heap Operations

    将N 个关键字以任意顺序放入树中，保持结构特性，再执行下滤

    -
    for (i = N/2; i > 0; i--)
- PercolateDown(i);
-
    +

    c + for (i = N/2; i > 0; i--) + PercolateDown(i);

    $$ T(N)=O(N) $$

    @@ -1696,31 +1696,31 @@

    4.4 Applications of Priority Queues<

    Heap Sort

    查找一个序列中第k小的元素

    The function is to find the K-th smallest element in a list A of N elements. The function BuildMaxHeap(H, K) is to arrange elements H[1] ... H[K] into a max-heap.

    -
    ElementType FindKthSmallest ( int A[], int N, int K )
-{   /* it is assumed that K<=N */
-    ElementType *H;
-    int i, next, child;
-
-    H = (ElementType*)malloc((K+1)*sizeof(ElementType));
-    for ( i = 1; i <= K; i++ ) H[i] = A[i-1];
-    BuildMaxHeap(H, K);
-
-    for ( next = K; next < N; next++ ) {
-        H[0] = A[next];
-        if ( H[0] < H[1] ) {
-            for ( i = 1; i*2 <= K; i = child ) {
-                child = i*2;
-                if ( child != K && H[child+1] > H[child] ) child++;
-                if ( H[0] < H[child] )
-                    H[i] = H[child];
-                else break;
-            }
-            H[i] = H[0];
-        }
-    }
-    return H[1];
-}
-
    +
    ElementType FindKthSmallest ( int A[], int N, int K )
+{   /* it is assumed that K<=N */
+    ElementType *H;
+    int i, next, child;
+
+    H = (ElementType*)malloc((K+1)*sizeof(ElementType));
+    for ( i = 1; i <= K; i++ ) H[i] = A[i-1];
+    BuildMaxHeap(H, K);
+
+    for ( next = K; next < N; next++ ) {
+        H[0] = A[next];
+        if ( H[0] < H[1] ) {
+            for ( i = 1; i*2 <= K; i = child ) {
+                child = i*2;
+                if ( child != K && H[child+1] > H[child] ) child++;
+                if ( H[0] < H[child] )
+                    H[i] = H[child];
+                else break;
+            }
+            H[i] = H[0];
+        }
+    }
+    return H[1];
+}
+

    4.5 \(d\)-Heaps — All nodes have \(d\) children

    Note :

    diff --git "a/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK07/index.html" "b/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK07/index.html" index 0af8b580..447555ab 100644 --- "a/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK07/index.html" +++ "b/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK07/index.html" @@ -1539,23 +1539,23 @@

    5.2 The Dynamic Equivalence Problem<
    • Given an equivalence relation ~, decide for any a and b if a ~ b
    -
    Algorithm: (Union/Find)
-{
-  /* step 1: read the relations in */
-    Initialize N disjoint sets;
-    while ( read in a ~ b )
-    {
-      if ( !(Find(a) == Find(b)) )  /*Dynamic(on-line)*/
-          Union the two sets;
-    } /* end-while */
-    /* step 2: decide if a ~ b */
-    while ( read in a and b )
-        if ( Find(a) == Find(b) )
-          output( true );
-        else   
-          output( false );
-}
-
    +

    pseudocode + Algorithm: (Union/Find) + { + /* step 1: read the relations in */ + Initialize N disjoint sets; + while ( read in a ~ b ) + { + if ( !(Find(a) == Find(b)) ) /*Dynamic(on-line)*/ + Union the two sets; + } /* end-while */ + /* step 2: decide if a ~ b */ + while ( read in a and b ) + if ( Find(a) == Find(b) ) + output( true ); + else + output( false ); + }

    • Elements of the sets : \(1,2,3,\cdots,N\)
    • Sets : \(S_1,S_2,\cdots\,and\,S_i\bigcap S_j=\emptyset\,(if\quad i\neq j)\)
      • @@ -1594,11 +1594,11 @@

      Union( \(i, j\) )

    • 数组初始化全部为0
    -
    void SetUnion(DisjSet S, SetType Rt1, SetType Rt2)
-{
-  S[Rt2] = Rt1;
-}
-
    +

    c + void SetUnion(DisjSet S, SetType Rt1, SetType Rt2) + { + S[Rt2] = Rt1; + }

    Find( \(i\) )

    • Implementation 1 :
      • @@ -1607,18 +1607,18 @@

      Find( \(i\) )

      • Implementation 2 :
      -
      SetType Find(ElementType X, DisjSet S)
-{
-  for ( ; S[X]>0; X=S[X]);
-  return X;
-}
-
      +

      c + SetType Find(ElementType X, DisjSet S) + { + for ( ; S[X]>0; X=S[X]); + return X; + }

      Analysis

      • Union and find are always paired. Thus we consider the performance of a sequence of union-find operations.

      7-3

      -
      Algorithm using union-find operations:
+
      Algorithm using union-find operations:
 {  
     Initialize Si = { i }  for  i = 1, ..., 12 ;
     for ( k = 1; k <= 9; k++ )  /* for each pair i R j */
@@ -1627,7 +1627,7 @@ 
      Analysis
      
             SetUnion( Find( i ), Find( j ) );
     }
 }
-
      +
      • Worst case : \(T(N)=\Theta(N^2)\)
      @@ -1649,64 +1649,64 @@

      Union-by-Size

      • Time complexity of \(N\) Union and \(M\) Find operations is now \(O(N+M\log_2N)\).
      -
      /* Assumes Rootl and Root2 are roots*/
-void SetUnion(DisjSet S, SetType Root1, SetType Root2)
-{
-    if (S[Root1] <= S[Root2])
-    {
-        S[Root1] += S[Root2];
-        S[Root2] = Root1;
-    }
-    else
-    {
-        S[Root2] += S[Root1];
-        S[Root1] = Root2;
-    }
-}
-
      +
      /* Assumes Rootl and Root2 are roots*/
+void SetUnion(DisjSet S, SetType Root1, SetType Root2)
+{
+    if (S[Root1] <= S[Root2])
+    {
+        S[Root1] += S[Root2];
+        S[Root2] = Root1;
+    }
+    else
+    {
+        S[Root2] += S[Root1];
+        S[Root1] = Root2;
+    }
+}
+

      Union-by-Height

      • Always change the shallow tree
      • 保证所有的树的深度最多是\(O(logN)\)
      -
      /* Assumes Rootl and Root2 are roots*/
-void SetUnion(DisjSet S, SetType Root1, SetType Root2)
-{
-    if ( S[Root2] < S[Root1])  /*Root2 is deeper set*/
-        S[Root1] = Root2;      /*Make Root2 new root*/
-    else
-    {
-        if (S[Root1] == S[Root2])  /*Same height*/
-            S[Root1]--;
-        S[Root2] = Root1;
-    }
-}
-
      +
      /* Assumes Rootl and Root2 are roots*/
+void SetUnion(DisjSet S, SetType Root1, SetType Root2)
+{
+    if ( S[Root2] < S[Root1])  /*Root2 is deeper set*/
+        S[Root1] = Root2;      /*Make Root2 new root*/
+    else
+    {
+        if (S[Root1] == S[Root2])  /*Same height*/
+            S[Root1]--;
+        S[Root2] = Root1;
+    }
+}
+

      5.5 Path Compression

      • 从X到Root的路径上的每一个结点都使它的父结点变成Root
      -
      SetType Find( ElementType X, DisjSet S )
-{
-    if ( S[ X ] <= 0 )    
-        return X;
-    else 
-        return S[ X ] = Find( S[ X ], S );
-}
-
      -
      SetType Find( ElementType X, DisjSet S )
-{   
-    ElementType root, trail, lead;
-    for ( root = X; S[ root ] > 0; root = S[ root ] );  /* find the root */
-    for ( trail = X; trail != root; trail = lead )
-    {
-        lead = S[ trail ];   
-        S[ trail ] = root;   
-    }  /* collapsing */
-    return root;
-}
-
      +
      SetType Find( ElementType X, DisjSet S )
+{
+    if ( S[ X ] <= 0 )    
+        return X;
+    else 
+        return S[ X ] = Find( S[ X ], S );
+}
+
      +
      SetType Find( ElementType X, DisjSet S )
+{   
+    ElementType root, trail, lead;
+    for ( root = X; S[ root ] > 0; root = S[ root ] );  /* find the root */
+    for ( trail = X; trail != root; trail = lead )
+    {
+        lead = S[ trail ];   
+        S[ trail ] = root;   
+    }  /* collapsing */
+    return root;
+}
+
      • Note : Not compatible with union-by-height since it changes the heights. Just take “height” as an estimated rank.
      diff --git "a/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK08/index.html" "b/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK08/index.html" index e84f17c8..18aeb4cf 100644 --- "a/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK08/index.html" +++ "b/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK08/index.html" @@ -1741,50 +1741,50 @@ 

      /*Test an AOV for feasibility, and generate a topological order if possible*/
-void Topsort( Graph G )
-{   
-    int Counter;
-    Vertex V, W;
-    for ( Counter = 0; Counter < NumVertex; Counter++ ) 
-    {
-        V = FindNewVertexOfDegreeZero( );
-        if ( V == NotAVertex ) //确认图中无环？
-        {
-            Error ( “Graph has a cycle” );   
-            break;  
-        }
-        TopNum[ V ] = Counter; /* or output V */
-        for ( each W adjacent to V )
-            Indegree[ W ]––; //对所有前置选课要求有V的课程，在修之前必须修完的课少了一门
-    }
-}
-
      +
      /*Test an AOV for feasibility, and generate a topological order if possible*/
+void Topsort( Graph G )
+{   
+    int Counter;
+    Vertex V, W;
+    for ( Counter = 0; Counter < NumVertex; Counter++ ) 
+    {
+        V = FindNewVertexOfDegreeZero( );
+        if ( V == NotAVertex ) //确认图中无环？
+        {
+            Error ( “Graph has a cycle” );   
+            break;  
+        }
+        TopNum[ V ] = Counter; /* or output V */
+        for ( each W adjacent to V )
+            Indegree[ W ]––; //对所有前置选课要求有V的课程，在修之前必须修完的课少了一门
+    }
+}
+
      \[ T=O(|V|^2+|E|) = O(|V|^2) \]
      -
      /*Improvment:Keep all the unassigned vertices of degree 0 in a special box (queue or stack)*/
-void Topsort( Graph G )
-{   
-    Queue Q;
-    int Counter = 0;
-    Vertex V, W;
-    Q = CreateQueue( NumVertex );  
-    MakeEmpty( Q );
-    for ( each vertex V )
-        if ( Indegree[ V ] == 0 ) Enqueue( V, Q );
-    while ( !IsEmpty( Q ) ) 
-    {
-        V = Dequeue( Q );
-        TopNum[ V ] = ++Counter; /* assign next */
-        for ( each W adjacent to V )
-            if (––Indegree[ W ] == 0 ) Enqueue( W, Q );
-    }  /* end-while */
-    if ( Counter != NumVertex )
-    Error( “Graph has a cycle” );
-    DisposeQueue( Q ); /* free memory */
-}
-
      +
      /*Improvment:Keep all the unassigned vertices of degree 0 in a special box (queue or stack)*/
+void Topsort( Graph G )
+{   
+    Queue Q;
+    int Counter = 0;
+    Vertex V, W;
+    Q = CreateQueue( NumVertex );  
+    MakeEmpty( Q );
+    for ( each vertex V )
+        if ( Indegree[ V ] == 0 ) Enqueue( V, Q );
+    while ( !IsEmpty( Q ) ) 
+    {
+        V = Dequeue( Q );
+        TopNum[ V ] = ++Counter; /* assign next */
+        for ( each W adjacent to V )
+            if (––Indegree[ W ] == 0 ) Enqueue( W, Q );
+    }  /* end-while */
+    if ( Counter != NumVertex )
+    Error( “Graph has a cycle” );
+    DisposeQueue( Q ); /* free memory */
+}
+
      \[ T=O(|V|+|E|) \]
      diff --git "a/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK09/index.html" "b/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK09/index.html" index fd25fca5..07183364 100644 --- "a/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK09/index.html" +++ "b/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK09/index.html" @@ -1494,55 +1494,55 @@

      Unweighted Shortest Path
    • Table[ i ].Known ::= 1 if \(v_i\) is checked; or 0 if not
    • Table[ i ].Path ::= for tracking the path / initialized to be 0 /
    -
    void Unweighted( Table T )
-{   
-    int CurrDist;
-    Vertex V, W;
-    for ( CurrDist = 0; CurrDist < NumVertex; CurrDist++ ) 
-    {
-        for ( each vertex V )
-            if ( !T[ V ].Known && T[ V ].Dist == CurrDist ) 
-            {
-                T[ V ].Known = true;
-                for ( each W adjacent to V )
-                    if ( T[ W ].Dist == Infinity ) 
-                    {
-                        T[ W ].Dist = CurrDist + 1;
-                        T[ W ].Path = V;
-                    } /* end-if Dist == Infinity */
-            } /* end-if !Known && Dist == CurrDist */
-    }  /* end-for CurrDist */
-}
-
    +
    void Unweighted( Table T )
+{   
+    int CurrDist;
+    Vertex V, W;
+    for ( CurrDist = 0; CurrDist < NumVertex; CurrDist++ ) 
+    {
+        for ( each vertex V )
+            if ( !T[ V ].Known && T[ V ].Dist == CurrDist ) 
+            {
+                T[ V ].Known = true;
+                for ( each W adjacent to V )
+                    if ( T[ W ].Dist == Infinity ) 
+                    {
+                        T[ W ].Dist = CurrDist + 1;
+                        T[ W ].Path = V;
+                    } /* end-if Dist == Infinity */
+            } /* end-if !Known && Dist == CurrDist */
+    }  /* end-for CurrDist */
+}
+

    The worst case :

    9-1 $$ T(N)=O(|V|^2) $$ Improvement :

    -
    void Unweighted( Table T )
-{   
-    /* T is initialized with the source vertex S given */
-    Queue Q;
-    Vertex V, W;
-    Q = CreateQueue( NumVertex );
-    MakeEmpty( Q );
-    Enqueue( S, Q ); /* Enqueue the source vertex */
-    while ( !IsEmpty( Q ) ) 
-    {
-        V = Dequeue( Q );
-        T[ V ].Known = true; /* not really necessary */
-        for ( each W adjacent to V )
-            if ( T[ W ].Dist == Infinity ) 
-            {
-                T[ W ].Dist = T[ V ].Dist + 1;
-                T[ W ].Path = V;
-                Enqueue( W, Q );
-            } /* end-if Dist == Infinity */
-    } /* end-while */
-    DisposeQueue( Q ); /* free memory */
-}
-
    +
    void Unweighted( Table T )
+{   
+    /* T is initialized with the source vertex S given */
+    Queue Q;
+    Vertex V, W;
+    Q = CreateQueue( NumVertex );
+    MakeEmpty( Q );
+    Enqueue( S, Q ); /* Enqueue the source vertex */
+    while ( !IsEmpty( Q ) ) 
+    {
+        V = Dequeue( Q );
+        T[ V ].Known = true; /* not really necessary */
+        for ( each W adjacent to V )
+            if ( T[ W ].Dist == Infinity ) 
+            {
+                T[ W ].Dist = T[ V ].Dist + 1;
+                T[ W ].Path = V;
+                Enqueue( W, Q );
+            } /* end-if Dist == Infinity */
+    } /* end-while */
+    DisposeQueue( Q ); /* free memory */
+}
+
    \[ T=O(|V|+|E|) \]
    @@ -1555,60 +1555,60 @@
    Dijkstra’s Algorithm
  • Greedy Method : \(u\) is chosen so that distance[ u ] = min{ \(w \notin S\) | distance[ w ] } (If \(u\) is not unique, then we may select any of them)
  • if distance[\(u_1\)] < distance[\(u_2\)] and add \(u_1\) into \(S\), then distance [ \(u_2\) ] may change. If so, a shorter path from \(s\) to \(u_2\) must go through \(u_1\) and distance [ \(u_2\) ] = distance [ \(u_1\) ] + length(< \(u_1\), \(u_2\)>).
-
typedef int Vertex;
-struct TableEntry
-{
-    List Header; /*Adjacency list*/
-    int Known;
-    DistType Dist;
-    Vertex Path;
-};
-/*Vertices are numbered from 0*/
-#define NotAVertex (-1)
-typedef struct TableEntry Table[ NumVertex ];
-
-
void InitTable(Vertex Start, Graph G, Table T)
-{ 
-    int i;
-    ReadGraph(G, T); /* Read graph somehow */
-    for(i = 0; i < NumVertex; i++)
-    {
-        T[ i ].Known = False;
-        T[ i ].Dist = Infinity;
-        T[ i ].Path = NotAVertex;
-    }
-    T[ Start ].dist = O;
-}
-
-
/*Print shortest path to V after Dijkstra has run*/
-/*Assume that the path exists*/
-void PrintPath(Vertex V, Table T)
-{
-    if (T[ V ].Path != NotAVertex)
-    {
-        PrintPath(T[ V ].Path, T);
-        printf(" to") ;
-    }
-    printf("%v", V) ; /* %v is pseudocode * /
-
-
void Dijkstra( Table T )
-{ 
-    Vertex V, W;
-    for ( ; ; ) 
-    {
-        V = smallest unknown distance vertex;
-        if ( V == NotAVertex ) break; 
-        T[ V ].Known = true;
-        for ( each W adjacent to V )
-            if ( !T[ W ].Known ) 
-                if ( T[ V ].Dist + Cvw < T[ W ].Dist ) 
-                {
-                    Decrease( T[ W ].Dist to T[ V ].Dist + Cvw );
-                    T[ W ].Path = V;
-                } /* end-if update W */
-    } /* end-for( ; ; ) */
-}
-
+
typedef int Vertex;
+struct TableEntry
+{
+    List Header; /*Adjacency list*/
+    int Known;
+    DistType Dist;
+    Vertex Path;
+};
+/*Vertices are numbered from 0*/
+#define NotAVertex (-1)
+typedef struct TableEntry Table[ NumVertex ];
+
+
void InitTable(Vertex Start, Graph G, Table T)
+{ 
+    int i;
+    ReadGraph(G, T); /* Read graph somehow */
+    for(i = 0; i < NumVertex; i++)
+    {
+        T[ i ].Known = False;
+        T[ i ].Dist = Infinity;
+        T[ i ].Path = NotAVertex;
+    }
+    T[ Start ].dist = O;
+}
+
+
/*Print shortest path to V after Dijkstra has run*/
+/*Assume that the path exists*/
+void PrintPath(Vertex V, Table T)
+{
+    if (T[ V ].Path != NotAVertex)
+    {
+        PrintPath(T[ V ].Path, T);
+        printf(" to") ;
+    }
+    printf("%v", V) ; /* %v is pseudocode * /
+
+
void Dijkstra( Table T )
+{ 
+    Vertex V, W;
+    for ( ; ; ) 
+    {
+        V = smallest unknown distance vertex;
+        if ( V == NotAVertex ) break; 
+        T[ V ].Known = true;
+        for ( each W adjacent to V )
+            if ( !T[ W ].Known ) 
+                if ( T[ V ].Dist + Cvw < T[ W ].Dist ) 
+                {
+                    Decrease( T[ W ].Dist to T[ V ].Dist + Cvw );
+                    T[ W ].Path = V;
+                } /* end-if update W */
+    } /* end-for( ; ; ) */
+}
+
Implementation 1
  • Simply scan the table to find the smallest unknown distance vertex.——\(O(|V|)\)
    • @@ -1647,28 +1647,28 @@
    Improvements
  • Fibonacci heap
Graphs with Negative Edge Costs
-
void WeightedNegative( Table T )
-{
-    Queue Q;
-    Vertex V, W;
-    Q = CreateQueue (NumVertex );  
-    MakeEmpty( Q );
-    Enqueue( S, Q ); /*Enqueue the source vertex*/
-    while ( !IsEmpty( Q ) ) 
-    {
-        V = Dequeue( Q );
-        for ( each W adjacent to V )
-        if ( T[ V ].Dist + Cvw < T[ W ].Dist ) 
-        {
-            T[ W ].Dist = T[ V ].Dist + Cvw;
-            T[ W ].Path = V;
-            if ( W is not already in Q )
-                Enqueue( W, Q );
-        } /*end-if update*/
-    } /*end-while */
-    DisposeQueue( Q ); /*free memory*/
-}
-
+
void WeightedNegative( Table T )
+{
+    Queue Q;
+    Vertex V, W;
+    Q = CreateQueue (NumVertex );  
+    MakeEmpty( Q );
+    Enqueue( S, Q ); /*Enqueue the source vertex*/
+    while ( !IsEmpty( Q ) ) 
+    {
+        V = Dequeue( Q );
+        for ( each W adjacent to V )
+        if ( T[ V ].Dist + Cvw < T[ W ].Dist ) 
+        {
+            T[ W ].Dist = T[ V ].Dist + Cvw;
+            T[ W ].Path = V;
+            if ( W is not already in Q )
+                Enqueue( W, Q );
+        } /*end-if update*/
+    } /*end-while */
+    DisposeQueue( Q ); /*free memory*/
+}
+

Note : Negative-cost cycle will cause indefinite loop

diff --git "a/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK10/index.html" "b/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK10/index.html" index fda0794e..a7fa9bb9 100644 --- "a/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK10/index.html" +++ "b/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK10/index.html" @@ -1520,50 +1520,51 @@
Greedy Method

  • 连续地按照最小的权选择边,，并且当所选的边不产生环时就把它作为取定的边

    -
    void Kruskal( Graph G )
-{   
-   T = { };
-    while ( T contains less than |V|-1 edges && E is not empty ) 
+
    pseudocode
+ void Kruskal( Graph G )
+ {   
+    T = { };
+     while ( T contains less than |V|-1 edges && E is not empty ) 
+     {
+         choose a least cost edge (v, w) from E;  /*DeleteMin*/
+         delete (v, w) from E;
+         if ( (v, w) does not create a cycle in T )     
+            add (v, w) to T;  /*Union/Find*/
+         else     
+            discard (v, w);
+     }
+     if ( T contains fewer than |V|-1 edges )
+        Error( “No spanning tree” );
+ }
    
+
    ```c
+ void Kruskal(Graph G)
+ {
+    int EdgesAccepted;
+    DisjSet S;
+    PriorityQueue H;
+    Vertex U, V;
+    SetType Uset, Vset;
+    Edge E;
    
+
    Initialize(S);
+ReadGraphIntoHeapArray(G, H);
+BuildHeap(H);
+
+EdgesAccepted = 0;
+while(EdgesAccepted < NumVertex-1)
+{
+    E = DeleteMin(H); /*E = (U,V)*/
+    Uset = Find(U, S);
+    Vset = Find(V, S);
+    if(Uset != Vset)
     {
-        choose a least cost edge (v, w) from E;  /*DeleteMin*/
-        delete (v, w) from E;
-        if ( (v, w) does not create a cycle in T )     
-           add (v, w) to T;  /*Union/Find*/
-        else     
-           discard (v, w);
+        /*Accept the edge*/
+        EdgesAccepted++;
+        SetUnion(S, USet, VSet);
     }
-    if ( T contains fewer than |V|-1 edges )
-       Error( “No spanning tree” );
 }
-
    -
    void Kruskal(Graph G)
-{
-   int EdgesAccepted;
-   DisjSet S;
-   PriorityQueue H;
-   Vertex U, V;
-   SetType Uset, Vset;
-   Edge E;
-
-   Initialize(S);
-   ReadGraphIntoHeapArray(G, H);
-   BuildHeap(H);
-
-   EdgesAccepted = 0;
-   while(EdgesAccepted < NumVertex-1)
-   {
-       E = DeleteMin(H); /*E = (U,V)*/
-       Uset = Find(U, S);
-       Vset = Find(V, S);
-       if(Uset != Vset)
-       {
-           /*Accept the edge*/
-           EdgesAccepted++;
-           SetUnion(S, USet, VSet);
-       }
-   }
-}
-
    +     +

    } + ```

    $$ T=O(|E|\log|E|) $$

    diff --git "a/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK11/index.html" "b/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK11/index.html" index 3857cb14..eb42b834 100644 --- "a/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK11/index.html" +++ "b/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK11/index.html" @@ -1462,25 +1462,25 @@

    WEEK 11

    -
    /*a generalization of preorder traversal*/
-void DFS(Vertex V)
-{   
-    visited[ V ] = true;  /*mark this vertex to avoid cycles*/
-    for ( each W adjacent to V )
-        if ( !visited[ W ] ) DFS( W );
-} /*T = O(|E|+|V|) as long as adjacency lists are used*/
-
    +
    /*a generalization of preorder traversal*/
+void DFS(Vertex V)
+{   
+    visited[ V ] = true;  /*mark this vertex to avoid cycles*/
+    for ( each W adjacent to V )
+        if ( !visited[ W ] ) DFS( W );
+} /*T = O(|E|+|V|) as long as adjacency lists are used*/
+

    Undirected Graphs

    -
    void ListComponents(Graph G)
-{   
-    for ( each V in G ) 
-        if ( !visited[ V ] ) 
-        {
-            DFS( V );
-            printf("\n");
-        }
-}
-
    +
    void ListComponents(Graph G)
+{   
+    for ( each V in G ) 
+        if ( !visited[ V ] ) 
+        {
+            DFS( V );
+            printf("\n");
+        }
+}
+

    Biconnectivity

    • \(v\) is an articulation point if \(G'=DeleteVertex(G, v)\) has at least 2 connected components.
      • @@ -1518,7 +1518,7 @@

      Biconnectivity

      Low(u)=\min{Num(u),\min{Low(w)|w\,is\,a\,child\,of\,u},\min{Num(w)|(u,w)\,is\,a\,back\,edge}} $$
    -
    /*Assign Num and compute Parents*/
+
    /*Assign Num and compute Parents*/
 void AssignNum(Vertex V)
 {
     Vertex W;
@@ -1531,8 +1531,8 @@ 
    Biconnectivity
    
             AssignNum(W);
         }
 }
-
    -
    /*Assign Low; also check for articulation points*/
+
    +
    /*Assign Low; also check for articulation points*/
 void AssignLow(Vertex V)
 {
     Vertex W;
@@ -1551,8 +1551,8 @@ 
    Biconnectivity
    
                 Low[V] ＝ Min(Low[V], Num[W]); /*Rule 2*/
     }
 }
-
    -
    void FindArt(Vertex V)
+
    +
    void FindArt(Vertex V)
 {
     Vertex W;
     Visited[V] = True;
@@ -1572,7 +1572,7 @@ 
    Biconnectivity
    
                 Low[V] = Min(Low[V], Num[W]); /*Rule 2*/
     }
 }
-
    +

    Euler Circuits

    [Proposition] An Euler circuit is possible only if the graph is connected and each vertex has an even degree.
    [Proposition] An Euler tour is possible if there are exactly two vertices having odd degree. One must start at one of the odd-degree vertices.
    @@ -1587,8 +1587,8 @@
    7 Sorting

    7.1 Preliminaries

    -
    void X_Sort (ElementType A[], int N)
-
    +
    void X_Sort (ElementType A[], int N)
+
    • N must be a legal integer.
    • Assume integer array for the sake of simplicity.
      • @@ -1597,21 +1597,21 @@

      7.1 Preliminaries

    7.2 Insertion Sort

    -
    void Insertion(ElementType A[], int N)
-{ 
-    int j, P; 
-    ElementType Tmp; 
-
-    for ( P = 1; P < N; P++ ) 
-    { 
-        Tmp = A[ P ];  /*the next coming card*/
-        for ( j = P; j > 0 && A[ j - 1 ] > Tmp; j-- ) 
-            A[ j ] = A[ j - 1 ]; 
-            /*shift sorted cards to provide a position for the new coming card*/
-        A[ j ] = Tmp;  /*place the new card at the proper position*/
-    }/*end for-P-loop*/
-}
-
    +
    void Insertion(ElementType A[], int N)
+{ 
+    int j, P; 
+    ElementType Tmp; 
+
+    for ( P = 1; P < N; P++ ) 
+    { 
+        Tmp = A[ P ];  /*the next coming card*/
+        for ( j = P; j > 0 && A[ j - 1 ] > Tmp; j-- ) 
+            A[ j ] = A[ j - 1 ]; 
+            /*shift sorted cards to provide a position for the new coming card*/
+        A[ j ] = Tmp;  /*place the new card at the proper position*/
+    }/*end for-P-loop*/
+}
+

    • The worst case : Input A[ ] is in reverse order diff --git "a/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK12/index.html" "b/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK12/index.html" index 1037652a..c7ea43d1 100644 --- "a/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK12/index.html" +++ "b/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK12/index.html" @@ -1446,23 +1446,23 @@

      Shell’s Increment Sequence

      \[ h_t=\lfloor N/2\rfloor,h_k=\lfloor h_{k+1}/2\rfloor \]
      -
      void Shellsort( ElementType A[ ], int N ) 
-{ 
-    int i, j, Increment; 
-    ElementType Tmp; 
-    for ( Increment = N / 2; Increment > 0; Increment /= 2 )  /*h sequence */
-        for ( i = Increment; i < N; i++ ) 
-        { /* insertion sort */
-            Tmp = A[ i ]; 
-            for ( j = i; j >= Increment; j -= Increment ) 
-                if( Tmp < A[ j-Increment ] )
-                    A[ j ] = A[ j-Increment ]; 
-                else 
-                    break; 
-            A[ j ] = Tmp;
-        } /* end for-I and for-Increment loops */
-}
-
      +
      void Shellsort( ElementType A[ ], int N ) 
+{ 
+    int i, j, Increment; 
+    ElementType Tmp; 
+    for ( Increment = N / 2; Increment > 0; Increment /= 2 )  /*h sequence */
+        for ( i = Increment; i < N; i++ ) 
+        { /* insertion sort */
+            Tmp = A[ i ]; 
+            for ( j = i; j >= Increment; j -= Increment ) 
+                if( Tmp < A[ j-Increment ] )
+                    A[ j ] = A[ j-Increment ]; 
+                else 
+                    break; 
+            A[ j ] = Tmp;
+        } /* end for-I and for-Increment loops */
+}
+
      • [Theorem] The worst-case running time of Shellsort, using Shell’s increments, is \(\Theta( N^2 )\).
      @@ -1487,15 +1487,15 @@

      Conclusion

      7.5 Heapsort

      Algorithm1

      -
      void Heapsort( int N ) 
-{
-    BuildHeap( H );
-    for ( i = 0; i < N; i++ ) 
-        TmpH[ i ] = DeleteMin( H );
-    for ( i = 0; i < N; i++ ) 
-        H[ i ] = TmpH[ i ];
-}
-
      +
      void Heapsort( int N ) 
+{
+    BuildHeap( H );
+    for ( i = 0; i < N; i++ ) 
+        TmpH[ i ] = DeleteMin( H );
+    for ( i = 0; i < N; i++ ) 
+        H[ i ] = TmpH[ i ];
+}
+
      \[ T(N)=O(N\log N) \]
      @@ -1503,18 +1503,18 @@

      Algorithm1

    • The space requirement is doubled.

    Algorithm2

    -
    void Heapsort( ElementType A[ ], int N ) 
-{
-    int i; 
-    for ( i = N / 2; i >= 0; i-- ) /*BuildHeap*/ 
-        PercDown( A, i, N );
-    for ( i = N - 1; i > 0; i-- ) 
-    { 
-        Swap( &A[ 0 ], &A[ i ] ); /*DeleteMax*/ 
-        PercDown( A, 0, i ); 
-    } 
-}
-
    +
    void Heapsort( ElementType A[ ], int N ) 
+{
+    int i; 
+    for ( i = N / 2; i >= 0; i-- ) /*BuildHeap*/ 
+        PercDown( A, i, N );
+    for ( i = N - 1; i > 0; i-- ) 
+    { 
+        Swap( &A[ 0 ], &A[ i ] ); /*DeleteMax*/ 
+        PercDown( A, 0, i ); 
+    } 
+}
+
    • [Theorem] The average number of comparisons used to heapsort a random permutation of N distinct items is \(2N\log N-O(N\log\log N)\).
    @@ -1523,54 +1523,54 @@

    Algorithm2

    7.6 Mergesort

    -
    void MSort( ElementType A[ ], ElementType TmpArray[ ], int Left, int Right ) 
-{   
-    int Center; 
-    if ( Left < Right ) 
-    {  /*if there are elements to be sort*/
-        Center = (Left+Right)/2; 
-        MSort(A, TmpArray, Left, Center);   /*T(N/2)*/
-        MSort(A, TmpArray, Center+1, Right);    /*T(N/2)*/
-        Merge(A, TmpArray, Left, Center+1, Right);  /*O(N)*/
-    } 
-} 
-
-void Mergesort( ElementType A[ ], int N ) 
-{   
-    ElementType *TmpArray;  /*need O(N) extra space*/
-    TmpArray = malloc(N*sizeof(ElementType)); 
-    if (TmpArray != NULL) 
-    { 
-        MSort(A, TmpArray, 0, N-1); 
-        free(TmpArray); 
-    } 
-    else FatalError("No space for tmp array!!!"); 
-}
-
    +
    void MSort( ElementType A[ ], ElementType TmpArray[ ], int Left, int Right ) 
+{   
+    int Center; 
+    if ( Left < Right ) 
+    {  /*if there are elements to be sort*/
+        Center = (Left+Right)/2; 
+        MSort(A, TmpArray, Left, Center);   /*T(N/2)*/
+        MSort(A, TmpArray, Center+1, Right);    /*T(N/2)*/
+        Merge(A, TmpArray, Left, Center+1, Right);  /*O(N)*/
+    } 
+} 
+
+void Mergesort( ElementType A[ ], int N ) 
+{   
+    ElementType *TmpArray;  /*need O(N) extra space*/
+    TmpArray = malloc(N*sizeof(ElementType)); 
+    if (TmpArray != NULL) 
+    { 
+        MSort(A, TmpArray, 0, N-1); 
+        free(TmpArray); 
+    } 
+    else FatalError("No space for tmp array!!!"); 
+}
+
    • If a TmpArray is declared locally for each call of Merge, then \(S(N) = O(N\log N)\).
    -
    /*Lpos = start of left half, Rpos = start of right half*/ 
-void Merge( ElementType A[ ], ElementType TmpArray[ ], int Lpos, int Rpos, int RightEnd ) 
-{   
-    int i, LeftEnd, NumElements, TmpPos; 
-    LeftEnd = Rpos-1; 
-    TmpPos = Lpos; 
-    NumElements = RightEnd-Lpos+1; 
-    while( Lpos <= LeftEnd && Rpos <= RightEnd ) /*main loop*/ 
-        if ( A[ Lpos ] <= A[ Rpos ] ) 
-            TmpArray[ TmpPos++ ] = A[ Lpos++ ]; 
-        else 
-            TmpArray[ TmpPos++ ] = A[ Rpos++ ]; 
-    while( Lpos <= LeftEnd ) /*Copy rest of first half*/ 
-        TmpArray[ TmpPos++ ] = A[ Lpos++ ]; 
-    while( Rpos <= RightEnd ) /*Copy rest of second half*/ 
-        TmpArray[ TmpPos++ ] = A[ Rpos++ ]; 
-    for( i = 0; i < NumElements; i++, RightEnd-- ) 
-        /*Copy TmpArray back*/ 
-        A[ RightEnd ] = TmpArray[ RightEnd ]; 
-}
-
    +
    /*Lpos = start of left half, Rpos = start of right half*/ 
+void Merge( ElementType A[ ], ElementType TmpArray[ ], int Lpos, int Rpos, int RightEnd ) 
+{   
+    int i, LeftEnd, NumElements, TmpPos; 
+    LeftEnd = Rpos-1; 
+    TmpPos = Lpos; 
+    NumElements = RightEnd-Lpos+1; 
+    while( Lpos <= LeftEnd && Rpos <= RightEnd ) /*main loop*/ 
+        if ( A[ Lpos ] <= A[ Rpos ] ) 
+            TmpArray[ TmpPos++ ] = A[ Lpos++ ]; 
+        else 
+            TmpArray[ TmpPos++ ] = A[ Rpos++ ]; 
+    while( Lpos <= LeftEnd ) /*Copy rest of first half*/ 
+        TmpArray[ TmpPos++ ] = A[ Lpos++ ]; 
+    while( Rpos <= RightEnd ) /*Copy rest of second half*/ 
+        TmpArray[ TmpPos++ ] = A[ Rpos++ ]; 
+    for( i = 0; i < NumElements; i++, RightEnd-- ) 
+        /*Copy TmpArray back*/ 
+        A[ RightEnd ] = TmpArray[ RightEnd ]; 
+}
+

    Analysis

    \[ T(1)=O(1)\\ diff --git "a/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK13/index.html" "b/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK13/index.html" index 7475a71e..f9740108 100644 --- "a/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK13/index.html" +++ "b/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK13/index.html" @@ -1502,7 +1502,7 @@

    7.7 Quicksort

  • the fastest known sorting algorithm in practice

    • Algorithm

    -
    void Quicksort( ElementType A[ ], int N )
+
    void Quicksort( ElementType A[ ], int N )
 {
     if (N < 2) return;
     pivot = pick any element in A[ ]; 
@@ -1510,7 +1510,7 @@ 
    Algorithm
    
         A1 = { a in S | a <= pivot } and A2 = { a in S | a >= pivot };
     A = Quicksort(A1, N1) and { pivot } and Quicksort(A2, N2);
 }
-
    +
    • The pivot is placed at the right place once and for all.
    • 要研究的问题是如何选取枢纽元和如何划分
      • @@ -1546,56 +1546,56 @@

      Small Arrays

    • Cutoff when \(N\) gets small and use other efficient algorithms (such as insertion sort).

    Implementation

    -
    void Quicksort( ElementType A[ ], int N ) 
-{ 
-    Qsort( A, 0, N-1 ); 
-    /*A:the array*/
-    /*0:Left index*/
-    /*N–1:Right index*/
-}
-
    -
    /* Return median of Left, Center, and Right */ 
-/* Order these and hide the pivot */ 
-ElementType Median3( ElementType A[ ], int Left, int Right ) 
-{
-    int Center = ( Left+Right )/2; 
-    if ( A[ Left ] > A[ Center ] ) 
-        Swap( &A[ Left ], &A[ Center ] ); 
-    if ( A[ Left ] > A[ Right ] ) 
-        Swap( &A[ Left ], &A[ Right ] ); 
-    if ( A[ Center ] > A[ Right ] ) 
-        Swap( &A[ Center ], &A[ Right ] ); 
-    /*Invariant: A[ Left ] <= A[ Center ] <= A[ Right ]*/ 
-    Swap( &A[ Center ], &A[ Right-1 ] ); /*Hide pivot*/ 
-    /*only need to sort A[ Left+1 ] … A[ Right–2 ]*/
-    return A[ Right-1 ];  /*Return pivot*/ 
-}
-
    -
    void Qsort( ElementType A[ ], int Left, int Right ) 
-{
-    int i, j; 
-    ElementType Pivot; 
-    if ( Left + Cutoff <= Right ) 
-    {   /*if the sequence is not too short*/
-        Pivot = Median3( A, Left, Right );  /*select pivot*/
-        i = Left;     
-        j = Right – 1;  /*why not set Left+1 and Right-2?*/
-        for( ; ; ) 
-        { 
-            while ( A[ ++i ] < Pivot ) { }  /*scan from left*/
-            while ( A[ --j ] > Pivot ) { }  /*scan from right*/
-            if ( i < j ) 
-                Swap( &A[ i ], &A[ j ] );  /*adjust partition*/
-            else break;  /*partition done*/
-        } 
-        Swap( &A[ i ], &A[ Right-1 ] ); /*restore pivot */ 
-        Qsort( A, Left, i-1 );    /*recursively sort left part*/
-        Qsort( A, i+1, Right );   /*recursively sort right part*/
-    }  /*end if - the sequence is long*/
-    else /*do an insertion sort on the short subarray*/ 
-        InsertionSort( A+Left, Right-Left+1 );
-}
-
    +
    void Quicksort( ElementType A[ ], int N ) 
+{ 
+    Qsort( A, 0, N-1 ); 
+    /*A:the array*/
+    /*0:Left index*/
+    /*N–1:Right index*/
+}
+
    +
    /* Return median of Left, Center, and Right */ 
+/* Order these and hide the pivot */ 
+ElementType Median3( ElementType A[ ], int Left, int Right ) 
+{
+    int Center = ( Left+Right )/2; 
+    if ( A[ Left ] > A[ Center ] ) 
+        Swap( &A[ Left ], &A[ Center ] ); 
+    if ( A[ Left ] > A[ Right ] ) 
+        Swap( &A[ Left ], &A[ Right ] ); 
+    if ( A[ Center ] > A[ Right ] ) 
+        Swap( &A[ Center ], &A[ Right ] ); 
+    /*Invariant: A[ Left ] <= A[ Center ] <= A[ Right ]*/ 
+    Swap( &A[ Center ], &A[ Right-1 ] ); /*Hide pivot*/ 
+    /*only need to sort A[ Left+1 ] … A[ Right–2 ]*/
+    return A[ Right-1 ];  /*Return pivot*/ 
+}
+
    +
    void Qsort( ElementType A[ ], int Left, int Right ) 
+{
+    int i, j; 
+    ElementType Pivot; 
+    if ( Left + Cutoff <= Right ) 
+    {   /*if the sequence is not too short*/
+        Pivot = Median3( A, Left, Right );  /*select pivot*/
+        i = Left;     
+        j = Right – 1;  /*why not set Left+1 and Right-2?*/
+        for( ; ; ) 
+        { 
+            while ( A[ ++i ] < Pivot ) { }  /*scan from left*/
+            while ( A[ --j ] > Pivot ) { }  /*scan from right*/
+            if ( i < j ) 
+                Swap( &A[ i ], &A[ j ] );  /*adjust partition*/
+            else break;  /*partition done*/
+        } 
+        Swap( &A[ i ], &A[ Right-1 ] ); /*restore pivot */ 
+        Qsort( A, Left, i-1 );    /*recursively sort left part*/
+        Qsort( A, i+1, Right );   /*recursively sort right part*/
+    }  /*end if - the sequence is long*/
+    else /*do an insertion sort on the short subarray*/ 
+        InsertionSort( A+Left, Right-Left+1 );
+}
+

    Note : If set i = Left+1 and j = Right-2, there will be an infinite loop if A[i] = A[j] = pivot.

    @@ -1695,38 +1695,38 @@

    Quickselect

    • 查找第\(K\)最大(最小)元
    -
    /*Places the kth sma11est element in the kth position*/
-/*Because arrays start at 0, this will be index k-1*/
-void Qselect(ElementType A[ ], int k, int Left, int Right)
-{
-    int i, j;
-    ElementType Pivot;
-
-    if (Left + Cutoff <= Right)
-    {
-        Pivot = Median3(A, Left, Right);
-        i = Left; 
-        j = Right-1;
-        for( ; ; )
-        {
-            while(A[ ++i ] < Pivot){ }
-            while(A[ --j ] > Pivot){ }
-            if(i < j)
-                Swap(&A[ i ], &A[ j ]);
-            else
-                break;
-        }
-        Swap(&A[ i ], &A[ Right-1 ]); /*Restore pivot*/
-
-        if(k <= i)
-            Qselect(A, k, Left, i-1);
-        else if (k > i+1)
-            Qselect(A, k, i+1, Right);
-    }
-    else /*Doan insertion sort on the subarray*/
-        InsertionSort(A+Left, Right-Left+1);
-}
-
    +
    /*Places the kth sma11est element in the kth position*/
+/*Because arrays start at 0, this will be index k-1*/
+void Qselect(ElementType A[ ], int k, int Left, int Right)
+{
+    int i, j;
+    ElementType Pivot;
+
+    if (Left + Cutoff <= Right)
+    {
+        Pivot = Median3(A, Left, Right);
+        i = Left; 
+        j = Right-1;
+        for( ; ; )
+        {
+            while(A[ ++i ] < Pivot){ }
+            while(A[ --j ] > Pivot){ }
+            if(i < j)
+                Swap(&A[ i ], &A[ j ]);
+            else
+                break;
+        }
+        Swap(&A[ i ], &A[ Right-1 ]); /*Restore pivot*/
+
+        if(k <= i)
+            Qselect(A, k, Left, i-1);
+        else if (k > i+1)
+            Qselect(A, k, i+1, Right);
+    }
+    else /*Doan insertion sort on the subarray*/
+        InsertionSort(A+Left, Right-Left+1);
+}
+

    image-20210125115951471

    正确答案是D

    @@ -1749,7 +1749,7 @@

    7.10 Bucket Sort

    image-20201221203533117

    -
    Algorithm
+
    Algorithm
 {
     initialize count[ ];
     while(read in a student’s record)
@@ -1759,7 +1759,7 @@ 
    7.10 Bucket Sort
    
         if(count[i]) output list count[i];
     }
 }
-
    +
    \[ T(N,M)=O(M+N) \]
    diff --git "a/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK14/index.html" "b/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK14/index.html" index dbb4e908..6358d1af 100644 --- "a/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK14/index.html" +++ "b/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK14/index.html" @@ -1473,80 +1473,80 @@

    8.3 Separate Chaining

    • keep a list of all keys that hash to the same value
    -
    struct ListNode; 
-typedef struct ListNode *Position; 
-struct HashTbl; 
-typedef struct HashTbl *HashTable; 
-struct ListNode { 
-    ElementType Element; 
-    Position Next; 
-}; 
-typedef Position List; 
-/* List *TheList will be an array of lists, allocated later */ 
-/* The lists use headers (for simplicity), */ 
-/* though this wastes space */ 
-struct HashTbl { 
-    int TableSize; 
-    List *TheLists; 
-}; 
-
    +
    struct ListNode; 
+typedef struct ListNode *Position; 
+struct HashTbl; 
+typedef struct HashTbl *HashTable; 
+struct ListNode { 
+    ElementType Element; 
+    Position Next; 
+}; 
+typedef Position List; 
+/* List *TheList will be an array of lists, allocated later */ 
+/* The lists use headers (for simplicity), */ 
+/* though this wastes space */ 
+struct HashTbl { 
+    int TableSize; 
+    List *TheLists; 
+}; 
+

    Create an empty table

    -
    HashTable InitializeTable( int TableSize ) 
-{   
-    HashTable H; 
-    int i; 
-    if ( TableSize < MinTableSize ) 
-    { 
-        Error( "Table size too small" );  
-        return NULL;  
-    } 
-    H = malloc( sizeof( struct HashTbl ) );  /*Allocate table*/
-    if ( H == NULL ) FatalError( "Out of space!!!" ); 
-    H->TableSize = NextPrime( TableSize );  /*Better be prime*/
-    H->TheLists = malloc( sizeof( List )* H->TableSize );  /*Array of lists*/
-    if ( H->TheLists == NULL ) FatalError( "Out of space!!!" );
-    H->TheList = malloc(H->TableSize*sizeof(struct ListNode));
-    for( i = 0; i < H->TableSize; i++ ) 
-    {   /*Allocate list headers*/
-        //H->TheLists[ i ] = malloc( sizeof( struct ListNode ) ); /* Slow! */
-        if ( H->TheLists[ i ] == NULL ) FatalError( "Out of space!!!" ); 
-        else H->TheLists[ i ]->Next = NULL;
-    } 
-    return H; 
-} 
-
    +
    HashTable InitializeTable( int TableSize ) 
+{   
+    HashTable H; 
+    int i; 
+    if ( TableSize < MinTableSize ) 
+    { 
+        Error( "Table size too small" );  
+        return NULL;  
+    } 
+    H = malloc( sizeof( struct HashTbl ) );  /*Allocate table*/
+    if ( H == NULL ) FatalError( "Out of space!!!" ); 
+    H->TableSize = NextPrime( TableSize );  /*Better be prime*/
+    H->TheLists = malloc( sizeof( List )* H->TableSize );  /*Array of lists*/
+    if ( H->TheLists == NULL ) FatalError( "Out of space!!!" );
+    H->TheList = malloc(H->TableSize*sizeof(struct ListNode));
+    for( i = 0; i < H->TableSize; i++ ) 
+    {   /*Allocate list headers*/
+        //H->TheLists[ i ] = malloc( sizeof( struct ListNode ) ); /* Slow! */
+        if ( H->TheLists[ i ] == NULL ) FatalError( "Out of space!!!" ); 
+        else H->TheLists[ i ]->Next = NULL;
+    } 
+    return H; 
+} 
+

    Find a key from a hash table

    -
    Position Find ( ElementType Key, HashTable H ) 
-{ 
-    Position P; 
-    List L; 
-    L = H->TheLists[ Hash( Key, H->TableSize ) ]; 
-    P = L->Next; 
-    while( P != NULL && P->Element != Key )  /*Probably need strcmp*/ 
-        P = P->Next; 
-    return P; 
-} 
-
    +
    Position Find ( ElementType Key, HashTable H ) 
+{ 
+    Position P; 
+    List L; 
+    L = H->TheLists[ Hash( Key, H->TableSize ) ]; 
+    P = L->Next; 
+    while( P != NULL && P->Element != Key )  /*Probably need strcmp*/ 
+        P = P->Next; 
+    return P; 
+} 
+

    Insert a key into a hash table

    -
    void Insert ( ElementType Key, HashTable H ) 
-{ 
-    Position Pos, NewCell; 
-    List L; 
-    Pos = Find( Key, H ); 
-    if ( Pos == NULL ) 
-    {   /*Key is not found, then insert*/
-        NewCell = malloc( sizeof( struct ListNode ) ); 
-        if ( NewCell == NULL ) FatalError( "Out of space!!!" ); 
-        else 
-        { 
-            L = H->TheLists[ Hash( Key, H->TableSize ) ]; /*Compute again is bad*/
-            NewCell->Next = L->Next; 
-            NewCell->Element = Key; /*Probably need strcpy!*/ 
-            L->Next = NewCell; 
-        } 
-    } 
-} 
-
    +
    void Insert ( ElementType Key, HashTable H ) 
+{ 
+    Position Pos, NewCell; 
+    List L; 
+    Pos = Find( Key, H ); 
+    if ( Pos == NULL ) 
+    {   /*Key is not found, then insert*/
+        NewCell = malloc( sizeof( struct ListNode ) ); 
+        if ( NewCell == NULL ) FatalError( "Out of space!!!" ); 
+        else 
+        { 
+            L = H->TheLists[ Hash( Key, H->TableSize ) ]; /*Compute again is bad*/
+            NewCell->Next = L->Next; 
+            NewCell->Element = Key; /*Probably need strcpy!*/ 
+            L->Next = NewCell; 
+        } 
+    } 
+} 
+

    Note : Make the TableSize about as large as the number of keys expected (i.e. to make the loading density factor \(\lambda\approx\)1).

    @@ -1555,7 +1555,7 @@

    8.4 Open Addressing

    • find another empty cell to solve collision(avoiding pointers)
    -
    Algorithm: insert key into an array of hash table
+
    Algorithm: insert key into an array of hash table
 {
     index = hash(key);
     initialize i = 0 ------ the counter of probing;
@@ -1568,7 +1568,7 @@ 
    8.4 Open Addressing
    
     if (table is full) ERROR (“No space left”);
     else insert key at index;
 }
-
    +

    Note : Generally \(\lambda<0.5\).

    diff --git "a/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK15/index.html" "b/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK15/index.html" index 24eb6bb0..cee35ec0 100644 --- "a/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK15/index.html" +++ "b/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/DS WEEK15/index.html" @@ -1371,58 +1371,58 @@
    \(4k + 3\), then the quadratic probing \(f(i) = \pm i^2\) can probe the entire table.

    -
    HashTable InitializeTable(int TableSize)
-{
-    HashTable H;
-    int i;
-    if(TableSize < MinTableSize)
-    {
-        Error("Table size too small");
-        return NULL;
-    }
-    /*Allocate table*/
-    H = malloc(sizeof(struct HashTbl));
-    if(H == NULL)
-        Fatal Error("Out of space!!!");
-    H->TableSize = NextPrime(TableSize);
-
-    /*Allocate array of Cells*/
-    H->TheCells = malloc(sizeof(Cell)*H->TableSize);
-    if(H->TheCells == NULL)
-        FatalError("Out of space!!!");
-
-    for(i = 0; i < H->TableSize; i++)
-        H->TheCells[ i ].Info = Empty;
-    return H;
-}
-
    -
    Position Find(ElementType Key, HashTable H) 
-{   
-    Position CurrentPos; 
-    int CollisionNum; 
-    CollisionNum = 0; 
-    CurrentPos = Hash(Key, H->TableSize); 
-    while(H->TheCells[ CurrentPos ].Info != Empty &&
-          H->TheCells[ CurrentPos ].Element != Key) 
-    { 
-        CurrentPos += 2*++CollisionNum-1; 
-        if (CurrentPos >= H->TableSize)  
-            CurrentPos -= H->TableSize;   /*Faster than mod*/
-    } 
-    return CurrentPos; 
-} 
-
    -
    void Insert(ElementType Key, HashTable H) 
-{ 
-    Position Pos; 
-    Pos = Find(Key, H); 
-    if (H->TheCells[ Pos ].Info != Legitimate) 
-    { /*OK to insert here*/ 
-        H->TheCells[ Pos ].Info = Legitimate; 
-        H->TheCells[ Pos ].Element = Key; /*Probably need strcpy*/ 
-    } 
-} 
-
    +
    HashTable InitializeTable(int TableSize)
+{
+    HashTable H;
+    int i;
+    if(TableSize < MinTableSize)
+    {
+        Error("Table size too small");
+        return NULL;
+    }
+    /*Allocate table*/
+    H = malloc(sizeof(struct HashTbl));
+    if(H == NULL)
+        Fatal Error("Out of space!!!");
+    H->TableSize = NextPrime(TableSize);
+
+    /*Allocate array of Cells*/
+    H->TheCells = malloc(sizeof(Cell)*H->TableSize);
+    if(H->TheCells == NULL)
+        FatalError("Out of space!!!");
+
+    for(i = 0; i < H->TableSize; i++)
+        H->TheCells[ i ].Info = Empty;
+    return H;
+}
+
    +
    Position Find(ElementType Key, HashTable H) 
+{   
+    Position CurrentPos; 
+    int CollisionNum; 
+    CollisionNum = 0; 
+    CurrentPos = Hash(Key, H->TableSize); 
+    while(H->TheCells[ CurrentPos ].Info != Empty &&
+          H->TheCells[ CurrentPos ].Element != Key) 
+    { 
+        CurrentPos += 2*++CollisionNum-1; 
+        if (CurrentPos >= H->TableSize)  
+            CurrentPos -= H->TableSize;   /*Faster than mod*/
+    } 
+    return CurrentPos; 
+} 
+
    +
    void Insert(ElementType Key, HashTable H) 
+{ 
+    Position Pos; 
+    Pos = Find(Key, H); 
+    if (H->TheCells[ Pos ].Info != Legitimate) 
+    { /*OK to insert here*/ 
+        H->TheCells[ Pos ].Info = Legitimate; 
+        H->TheCells[ Pos ].Element = Key; /*Probably need strcpy*/ 
+    } 
+} 
+

    Note :

      @@ -1458,24 +1458,24 @@

      8.5 Rehashing

      Note : Usually there should have been N/2 insertions before rehash, so O(N) rehash only adds a constant cost to each insertion. However, in an interactive system, the unfortunate user whose insertion caused a rehash could see a slowdown.

      -
      HashTable Rehash(HashTable H)
-{
-    int i, OldSize;
-    Cell *OldCells;
-    OldCells = H->TheCells;
-    OldSize = H->TableSize;
-
-    /*Get a new, empty table*/
-    H = InitializeTable(2*OldSize);
-    /*Scan through old table, reinserting into new*/
-    for(i = 0; i < OldSize; i++)
-        if(OldCells[i].Info == Legitimate)
-            Insert(OldCells[i].Element, H);
-    free(OldCells);
-
-    return H;
-}
-
      +
      HashTable Rehash(HashTable H)
+{
+    int i, OldSize;
+    Cell *OldCells;
+    OldCells = H->TheCells;
+    OldSize = H->TableSize;
+
+    /*Get a new, empty table*/
+    H = InitializeTable(2*OldSize);
+    /*Scan through old table, reinserting into new*/
+    for(i = 0; i < OldSize; i++)
+        if(OldCells[i].Info == Legitimate)
+            Insert(OldCells[i].Element, H);
+    free(OldCells);
+
+    return H;
+}
+
      diff --git "a/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/Foundation-of-Data-Structure/index.html" "b/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/Foundation-of-Data-Structure/index.html" index 388913d1..a177ea70 100644 --- "a/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/Foundation-of-Data-Structure/index.html" +++ "b/site/2 \346\225\260\346\215\256\347\273\223\346\236\204/webnotes/Foundation-of-Data-Structure/index.html" @@ -1206,9 +1206,9 @@
      • - + - [Definition] Data Type = {Objects} and {Operations} + [Definition] Data Type = {Objects} and @@ -3399,11 +3399,11 @@

        [Example] Selection Sort : Sort a set of \(n\geq1\) integers in increasing order

        -
        for (i = 0; i < n; i++){
+
        for (i = 0; i < n; i++){
     Examine list[i] to list[n-1] and suppose that the smallest integer is at list[min];
     Interchange list[i] and list[min];
 }
-
        +

        1.1 What to Analyze

          @@ -3432,17 +3432,17 @@

          1.1 What to Analyze

        • \(T_{avg}(N)\, and\, T_{worst}(N)\) : the average and worst case time complexities as functions of input size \(N\)

        [Example] Matrix addition

        -
        void add(int a[][MAX_SIZE],
-         int b[][MAX_SIZE],
-         int c[][MAX_SIZE],
-         int rows, int cols) 
-{
-    int i, j;
-    for (i=0; i<rows; i++)/*rows+1*/
-        for (j=0;j<cols;j++)/*rows(cols+1)*/
-            c[i][j] = a[i][j]+b[i][j];/*rows*cols*/
-}
-
        +
        void add(int a[][MAX_SIZE],
+         int b[][MAX_SIZE],
+         int c[][MAX_SIZE],
+         int rows, int cols) 
+{
+    int i, j;
+    for (i=0; i<rows; i++)/*rows+1*/
+        for (j=0;j<cols;j++)/*rows(cols+1)*/
+            c[i][j] = a[i][j]+b[i][j];/*rows*cols*/
+}
+
        \[ T(rows, cols) = 2rows\times cols + 2rows+1 \]
        @@ -3450,29 +3450,29 @@

        [Example] Matrix addition

      • 非对称

      [Example] Iterative function for summing a list of numbers

      -
      float sum (float list[], int n)
-{  /*add a list of numbers*/
-    float tempsum = 0; /*count = 1*/
-    int i;
-    for (i=0; i<n; i++)
-        /*count++*/
-        tempsum  += list[i]; /*count++*/
-    /*count++ for last excutaion of for*/
-   return tempsum; /*count++*/
-}
-
      +
      float sum (float list[], int n)
+{  /*add a list of numbers*/
+    float tempsum = 0; /*count = 1*/
+    int i;
+    for (i=0; i<n; i++)
+        /*count++*/
+        tempsum  += list[i]; /*count++*/
+    /*count++ for last excutaion of for*/
+   return tempsum; /*count++*/
+}
+
      \[ T_{sum}(n)=2n+3 \]

      [Example] Recursive function for summing a list of numbers

      -
      float rsum (float list[], int n)
-{/*add a list of numbers*/
-    if (n) /*count++*/
-        return rsum(list, n-1) + list[n-1];
-        /*count++*/
-    return 0; /*count++*/
-}
-
      +
      float rsum (float list[], int n)
+{/*add a list of numbers*/
+    if (n) /*count++*/
+        return rsum(list, n-1) + list[n-1];
+        /*count++*/
+    return 0; /*count++*/
+}
+
      \[ T_{rsum}(n)=2n+2 \]
      @@ -3515,17 +3515,17 @@

      [Definition]

      1-2

      [Example] Matrix addition

      -
      void add(int a[][MAX_SIZE],
-         int b[][MAX_SIZE],
-         int c[][MAX_SIZE],
-         int rows, int cols) 
-{
-    int i, j;
-    for (i=0; i<rows; i++)
-        for (j=0;j<cols;j++)
-            c[i][j] = a[i][j]+b[i][j];
-}
-
      +
      void add(int a[][MAX_SIZE],
+         int b[][MAX_SIZE],
+         int c[][MAX_SIZE],
+         int rows, int cols) 
+{
+    int i, j;
+    for (i=0; i<rows; i++)
+        for (j=0;j<cols;j++)
+            c[i][j] = a[i][j]+b[i][j];
+}
+
      \[ T(rows,cols)=\Theta(rows\cdot cols) \]
      @@ -3556,14 +3556,14 @@

      General Rules

      $$ Fib(0)=Fib(1)=1, Fib(n)=Fib(n-1)+Fib(n-2) $$

      -
      lont int Fib (int N) /*T(N)*/
-{
-    if (N<=1) /*O(1)*/
-        return 1; /*O(1)*/
-    else
-        return Fib(N-1)+Fib(N-2);
-}      /*O(1)*//*T(N-1)*//*T(N-2)*/
-
      +

      c + lont int Fib (int N) /*T(N)*/ + { + if (N<=1) /*O(1)*/ + return 1; /*O(1)*/ + else + return Fib(N-1)+Fib(N-2); + } /*O(1)*//*T(N-1)*//*T(N-2)*/

      $$ T(N)=T(N-1)+T(N-2)+2\geq Fib(N)\ \left(\frac{3}{2} \right)^n\leq Fib(N)\leq\left(\frac{5}{3}\right)^n @@ -3575,87 +3575,87 @@

      General Rules

      1.3 Compare the Algorithms

      [Example] 最大子序列和

      Algorithm 1

      -
      int  MaxSubsequenceSum ( const int A[ ],  int  N ) 
-{ 
-    int ThisSum, MaxSum, i, j, k; 
-    MaxSum = 0;   /* initialize the maximum sum */
-    for( i = 0; i < N; i++ )  /* start from A[ i ] */
-        for( j = i; j < N; j++ ) {   /* end at A[ j ] */
-            ThisSum = 0; 
-            for( k = i; k <= j; k++ ) 
-                ThisSum += A[ k ];  /* sum from A[ i ] to A[ j ] */
-            if ( ThisSum > MaxSum ) 
-                MaxSum = ThisSum;  /* update max sum */
-        }  /* end for-j and for-i */
-    return MaxSum; 
-}
-
      +
      int  MaxSubsequenceSum ( const int A[ ],  int  N ) 
+{ 
+    int ThisSum, MaxSum, i, j, k; 
+    MaxSum = 0;   /* initialize the maximum sum */
+    for( i = 0; i < N; i++ )  /* start from A[ i ] */
+        for( j = i; j < N; j++ ) {   /* end at A[ j ] */
+            ThisSum = 0; 
+            for( k = i; k <= j; k++ ) 
+                ThisSum += A[ k ];  /* sum from A[ i ] to A[ j ] */
+            if ( ThisSum > MaxSum ) 
+                MaxSum = ThisSum;  /* update max sum */
+        }  /* end for-j and for-i */
+    return MaxSum; 
+}
+
      \[ T(N)=O(N^3) \]

      Algotithm 2

      -
      int  MaxSubsequenceSum ( const int A[ ],  int  N ) 
-{ 
-    int ThisSum, MaxSum, i, j; 
-    MaxSum = 0;   /* initialize the maximum sum */
-    for( i = 0; i < N; i++ ) {   /* start from A[ i ] */
-        ThisSum = 0; 
-        for( j = i; j < N; j++ ) {   /* end at A[ j ] */
-            ThisSum += A[ j ];  /* sum from A[ i ] to A[ j ] */
-            if ( ThisSum > MaxSum ) 
-                MaxSum = ThisSum;  /* update max sum */
-        }  /* end for-j */
-    }  /* end for-i */
-    return MaxSum; 
-} 
-
      +
      int  MaxSubsequenceSum ( const int A[ ],  int  N ) 
+{ 
+    int ThisSum, MaxSum, i, j; 
+    MaxSum = 0;   /* initialize the maximum sum */
+    for( i = 0; i < N; i++ ) {   /* start from A[ i ] */
+        ThisSum = 0; 
+        for( j = i; j < N; j++ ) {   /* end at A[ j ] */
+            ThisSum += A[ j ];  /* sum from A[ i ] to A[ j ] */
+            if ( ThisSum > MaxSum ) 
+                MaxSum = ThisSum;  /* update max sum */
+        }  /* end for-j */
+    }  /* end for-i */
+    return MaxSum; 
+} 
+
      \[ T(N)=O(N^2) \]

      Algorithm 3 Divide and Conquer 分治法

      -
      static int MaxSubSum(const int A[ ], int Left, int Right)
-{
-    int MaxLeftSum, MaxRightSum;
-    int MaxLeftBorderSum, MaxRightBorderSum;
-    int LeftBorderSum, RightBorderSum;
-    int Center, i;
-
-    if (Left == Right)
-        if (A[Left] > 0)
-            return A[Left];
-        else
-            return 0;
-
-    Center = (Left + Right) / 2;
-    MaxLeftSum = MaxSubSum(A, Left, Center);
-    MaxRightSum = MaxSubSum(A, Center + 1, Right);
-
-    MaxLeftBorderSum = 0;
-    LeftBorderSum = 0;
-    for (i = Center; i >= Left; i--)
-    {
-        LeftBorderSum += A[i];
-        if (LeftBorderSum > MaxLeftBorderSum)
-            MaxLeftBorderSum = LeftBorderSum;
-    }
-
-    MaxRightBorderSum = 0;
-    RightBorderSum = 0;
-    for (i = Center+1; i <= Right; i++)
-    {
-        RightBorderSum += A[i];
-        if (RightBorderSum > MaxRightBorderSum)
-            MaxRightBorderSum = RightBorderSum;
-    }
-
-    return Max3(MaxLeftSum, MaxRightSum, MaxLeftBorderSum + MaxRightBorderSum);
-}
-
-int MaxSubsequenceSum(const int A[ ], int N)
-{
-    return MaxSubSum(A, 0, N - 1);
-}
-
      +
      static int MaxSubSum(const int A[ ], int Left, int Right)
+{
+    int MaxLeftSum, MaxRightSum;
+    int MaxLeftBorderSum, MaxRightBorderSum;
+    int LeftBorderSum, RightBorderSum;
+    int Center, i;
+
+    if (Left == Right)
+        if (A[Left] > 0)
+            return A[Left];
+        else
+            return 0;
+
+    Center = (Left + Right) / 2;
+    MaxLeftSum = MaxSubSum(A, Left, Center);
+    MaxRightSum = MaxSubSum(A, Center + 1, Right);
+
+    MaxLeftBorderSum = 0;
+    LeftBorderSum = 0;
+    for (i = Center; i >= Left; i--)
+    {
+        LeftBorderSum += A[i];
+        if (LeftBorderSum > MaxLeftBorderSum)
+            MaxLeftBorderSum = LeftBorderSum;
+    }
+
+    MaxRightBorderSum = 0;
+    RightBorderSum = 0;
+    for (i = Center+1; i <= Right; i++)
+    {
+        RightBorderSum += A[i];
+        if (RightBorderSum > MaxRightBorderSum)
+            MaxRightBorderSum = RightBorderSum;
+    }
+
+    return Max3(MaxLeftSum, MaxRightSum, MaxLeftBorderSum + MaxRightBorderSum);
+}
+
+int MaxSubsequenceSum(const int A[ ], int N)
+{
+    return MaxSubSum(A, 0, N - 1);
+}
+
      \[ \because T(N)=2T(\frac N2)+cN\quad T(1)=O(1)\\ T(\frac N2)=2T(\frac N {2^2})+c\frac N2\\ @@ -3664,20 +3664,20 @@

      [Example] 最大子序列和

      \therefore T(N)=2^kT(\frac N{2^k})+kcN=N\cdot O(1)+cN\log N \]

      Algorithm 4 On-line Algorithm 在线算法

      -
      int MaxSubsequenceSum( const int  A[ ],  int  N ) 
-{ 
-    int ThisSum, MaxSum, j; 
-    ThisSum = MaxSum = 0; 
-    for ( j = 0; j < N; j++ ) { 
-        ThisSum += A[ j ]; 
-        if ( ThisSum > MaxSum ) 
-            MaxSum = ThisSum; 
-        else if ( ThisSum < 0 ) 
-            ThisSum = 0;
-    }  /* end for-j */
-    return MaxSum; 
-} 
-
      +
      int MaxSubsequenceSum( const int  A[ ],  int  N ) 
+{ 
+    int ThisSum, MaxSum, j; 
+    ThisSum = MaxSum = 0; 
+    for ( j = 0; j < N; j++ ) { 
+        ThisSum += A[ j ]; 
+        if ( ThisSum > MaxSum ) 
+            MaxSum = ThisSum; 
+        else if ( ThisSum < 0 ) 
+            ThisSum = 0;
+    }  /* end for-j */
+    return MaxSum; 
+} 
+
      \[ T(N)=O(N) \]
      @@ -3691,49 +3691,49 @@

      1.4 Logrithms in the Running Time

    • 如果一个算法用常数时间将问题的大小削减为其一部分(通常是1/2)，那么该算法就是\(O(logN)\)的

    -
    int BinarySearch ( const ElementType A[ ], ElementType X, int N ) 
-{ 
-    int  Low, Mid, High; 
-    Low = 0;  High = N - 1; 
-    while ( Low <= High ) { 
-        Mid = ( Low + High ) / 2; 
-        if ( A[ Mid ] < X ) 
-            Low = Mid + 1; 
-        else 
-            if ( A[ Mid ] > X ) 
-                High = Mid - 1; 
-            else 
-                return  Mid; /* Found */ 
-    }  /* end while */
-    return  NotFound; /* NotFound is defined as -1 */ 
-} 
-
    +
    int BinarySearch ( const ElementType A[ ], ElementType X, int N ) 
+{ 
+    int  Low, Mid, High; 
+    Low = 0;  High = N - 1; 
+    while ( Low <= High ) { 
+        Mid = ( Low + High ) / 2; 
+        if ( A[ Mid ] < X ) 
+            Low = Mid + 1; 
+        else 
+            if ( A[ Mid ] > X ) 
+                High = Mid - 1; 
+            else 
+                return  Mid; /* Found */ 
+    }  /* end while */
+    return  NotFound; /* NotFound is defined as -1 */ 
+} 
+
    \[ T_{worst}(N)=O(\log N) \]

    [Example] Euclid’s Algorithm

    -
    int Gcd(int M, int N)
-{
-    int Rem;
-
-    while (N > 0)
-    {
-        Rem = M % N;
-        M = N;
-        N = Rem;
-    }
-    return M;
-}
-
    +
    int Gcd(int M, int N)
+{
+    int Rem;
+
+    while (N > 0)
+    {
+        Rem = M % N;
+        M = N;
+        N = Rem;
+    }
+    return M;
+}
+

    [Example] Efficient exponentiation

    -
    long int Pow(long int X, int N)
-{
-    if (N == 0) return 1;
-    if (N == 1) return X;
-    if (IsEven(N)) return Pow(X*X, N/2);/*return Pow(X, N/2)*Pow(X, N/2) affects the efficiency*/
-    else return Pow(X*X, N/2)*X; /*return Pow(X, N-1)*X is the same*/
-}
-
    +
    long int Pow(long int X, int N)
+{
+    if (N == 0) return 1;
+    if (N == 1) return X;
+    if (IsEven(N)) return Pow(X*X, N/2);/*return Pow(X, N/2)*Pow(X, N/2) affects the efficiency*/
+    else return Pow(X*X, N/2)*X; /*return Pow(X, N-1)*X is the same*/
+}
+

    1.5 Checking Your Analysis

    Method 1

    @@ -3745,7 +3745,7 @@

    Method 2

    2 LIst, Stacks and Queues

    2.1 Abstract Data Type(ADT) 抽象数据类型

    -

    [Definition] Data Type = {Objects} and {Operations}

    +

    [Definition] Data Type = {Objects} and

    [Definition] An Abstract Data Type(ADT) is a data type that is organized in such a way that the specification on the objects and specification of the operations on the objects are separated from the representation of the objects and the implementation on the operations.

    2.2 The List ADT

    @@ -3796,87 +3796,92 @@

    Linked Lists

    Finding take \(O(N)\) times.

    -
    /*Return true if L is empty*/
-int IsEmpty(List L)
-{
-  return L->Next == NULL;
-}
-
    -
    /*Return true if P is the last position in list L*/
-/*Parameter L is unused in this implementation*/
-int IsLast(Position P, List L)
-{
-  return P->Next == NULL;
-}
-
    -
    /*Return Position of X in L; NULL if not found*/
-Position Find(Element X, List L)
-{
-  Position P;
-
-  P = L->Next;
-  while (P != NULL && P->Element != X) P = P->Next;
-
-  return P;
-}
-
    -
    /*Delete first occurence of X from a list*/
-/*Assume use of a header node*/
-void Delete(ElementType X, List L)
-{
-  Position P, TmpCell;
-
-  P = FindPrevious(X, L);
-
-  if (!IsLast(P, L))
-  {
-      TmpCell = P->Next;
-      P->Next = TmpCell->Next;
-      free(TmpCell);
-  }
-}
-
    -
    /*If X is not found, then Next field of returned*/
-/*Assumes a header*/
-Position FindPrevious(ElementType X, List L)
-{
-  Position P;
-
-  P = L;
-  while (P->Next != NULL && P->Next->Element != X) P = P->Next;
-
-  return P;
-}
-
    -
    /*Insert (after legal position P)*/
-/*Header implementation assumed*/
-/*Parameter L is unused in this implementation*/
-void Insert(ElementType X, List L, Position P)
-{
-  Position TmpCell;
-
-  TmpCell = malloc(sizeof(struct Node));
-  if (TmpCell == NULL) FatalError("Out of space!")
-
-  TmpCell->Element = X;
-  TmpeCell->Next = P->Next;
-  P->Next = TmpCell;
-}
-
    -
    void DeleteList(List L)
-{
-  Position P, Tmp;
-
-  P = L->Next;
-  L->Next = NULL;
-  while (P != NULL)
-  {
-      Tmp = P->Next;
-      free(P);
-      P = Tmp;
-  }
-}
-
    +

    c + /*Return true if L is empty*/ + int IsEmpty(List L) + { + return L->Next == NULL; + }

    +

    c + /*Return true if P is the last position in list L*/ + /*Parameter L is unused in this implementation*/ + int IsLast(Position P, List L) + { + return P->Next == NULL; + }

    +

    ```c + /Return Position of X in L; NULL if not found/ + Position Find(Element X, List L) + { + Position P;

    +
    P = L->Next;
+while (P != NULL && P->Element != X) P = P->Next;
+
+return P;
+
    +

    } + ```

    +

    ```c + /Delete first occurence of X from a list/ + /Assume use of a header node/ + void Delete(ElementType X, List L) + { + Position P, TmpCell;

    +
    P = FindPrevious(X, L);
+
+if (!IsLast(P, L))
+{
+    TmpCell = P->Next;
+    P->Next = TmpCell->Next;
+    free(TmpCell);
+}
+
    +

    } + ```

    +

    ```c + /If X is not found, then Next field of returned/ + /Assumes a header/ + Position FindPrevious(ElementType X, List L) + { + Position P;

    +
    P = L;
+while (P->Next != NULL && P->Next->Element != X) P = P->Next;
+
+return P;
+
    +

    } + ```

    +

    ```c + /Insert (after legal position P)/ + /Header implementation assumed/ + /Parameter L is unused in this implementation/ + void Insert(ElementType X, List L, Position P) + { + Position TmpCell;

    +
    TmpCell = malloc(sizeof(struct Node));
+if (TmpCell == NULL) FatalError("Out of space!")
+
+TmpCell->Element = X;
+TmpeCell->Next = P->Next;
+P->Next = TmpCell;
+
    +

    } + ```

    +

    ```c + void DeleteList(List L) + { + Position P, Tmp;

    +
    P = L->Next;
+L->Next = NULL;
+while (P != NULL)
+{
+    Tmp = P->Next;
+    free(P);
+    P = Tmp;
+}
+
    +

    } + ```

    Doubly Linked Circular Lists

    • Finding take \(O(\frac N 2)\) times.
      • @@ -3909,58 +3914,60 @@

      Two Applications

      [Representation 1]

      -
      typedef struct {
-  int CoeffArray [ MaxDegree + 1 ] ;
-  int HighPower;
-}  *Polynomial ; 
-
      -
      /*将多项式初始化为零*/
-void ZeroPolynomial(Polynomial Poly)
-{
-  int i;
-  for(i = O; i <= MaxDegree; i++)
-      Poly->CoeffArray[ i ] = O;
-  Poly->HighPower = O;
-}
-
      -
      /*两个多项式相加*/
-void AddPolynomial(const Polynomial Poly1, const Polynomial Poly2, Polynomial PolySum)
-{
-    int i;
-
-    ZeroPolynomial(PolySum);
-  PolySum->HighPower = Max(Poly1->HighPower, Poly2->HighPower);
-
-    for (i = PolySum->HighPower; i >= O; i--)
-      PolySum->CoeffArray[ i ] = Poly1->CoeffArray[ i ] + Poly2->CoeffArray[ i ];
-}
-
      -
      void MultPolynomial(const Polynomial Poly1, const Polynomial Poly2, Polynomial PolyProd)
-{
-    int i, j;
-
-    ZeroPolynomial (PolyProd);
-  PolyProd->HighPower = Poly1->HighPower + Poly2->HighPower;
-
-    if(PolyProd->HighPower > MaxDegree)
-      Error("Exceeded array size");
-  else
-      for(i = O; i <= Poly1->HighPower; i++)
-          for(j = O; j <= Poly2->HighPower; j++)
-              PolyProd->CoeffArray[ i + j ] += Poly1->CoeffArray[ i ] * Poly2->CoeffArray[ j ];
-}
-
      +

      c + typedef struct { + int CoeffArray [ MaxDegree + 1 ] ; + int HighPower; + } *Polynomial ;

      +

      c + /*将多项式初始化为零*/ + void ZeroPolynomial(Polynomial Poly) + { + int i; + for(i = O; i <= MaxDegree; i++) + Poly->CoeffArray[ i ] = O; + Poly->HighPower = O; + }

      +

      ```c + /两个多项式相加/ + void AddPolynomial(const Polynomial Poly1, const Polynomial Poly2, Polynomial PolySum) + { + int i;

      +
        ZeroPolynomial(PolySum);
+PolySum->HighPower = Max(Poly1->HighPower, Poly2->HighPower);
+
+  for (i = PolySum->HighPower; i >= O; i--)
+    PolySum->CoeffArray[ i ] = Poly1->CoeffArray[ i ] + Poly2->CoeffArray[ i ];
+
      +

      } + ```

      +

      ```c + void MultPolynomial(const Polynomial Poly1, const Polynomial Poly2, Polynomial PolyProd) + { + int i, j;

      +
        ZeroPolynomial (PolyProd);
+PolyProd->HighPower = Poly1->HighPower + Poly2->HighPower;
+
+  if(PolyProd->HighPower > MaxDegree)
+    Error("Exceeded array size");
+else
+    for(i = O; i <= Poly1->HighPower; i++)
+        for(j = O; j <= Poly2->HighPower; j++)
+            PolyProd->CoeffArray[ i + j ] += Poly1->CoeffArray[ i ] * Poly2->CoeffArray[ j ];
+
      +

      } + ```

      • [Representation 2]
      -
      typedef struct poly_node *poly_ptr;
-struct poly_node{
-    int Coefficient;  /* assume coefficients are integers */
-    int Exponent;
-    poly_ptr Next;
-};
-typedef poly_ptr a;    /* nodes sorted by exponent */
-
      +

      c + typedef struct poly_node *poly_ptr; + struct poly_node{ + int Coefficient; /* assume coefficients are integers */ + int Exponent; + poly_ptr Next; + }; + typedef poly_ptr a; /* nodes sorted by exponent */

      • 只存储非零项

        @@ -3992,73 +3999,73 @@

        Linked List Implementatio
        • The calls to malloc and free are expensive. Simply keep another stack as a recycle bin.
        -
        int IsEmpty(Stack S)
-{
-  return S->Next == NULL;
-}
-
        -
        Stack CreateStack(void)
-{
-  Stack S;
-  S = malloc(sizeof(struct Node));
-  if (S == NULL)
-      Fatal Error("Out of space!");
-  S->Next == NULL;
-  MakeEmpty(S);
-  return S;
-}
-
-void MakeEmpty(Stack S)
-{
-  if (S == NULL)
-      Error("Must use CreateStack first");
-  else
-      while(!IsEmpty(S)) Pop(S);
-}
-
        -
        void Push(ElementType X, Stack S)
-{
-  PtrToNode TmpCell;
-  TmpCell = malloc(sizeof(struct Node));
-  if (TmpCell == NULL)
-      Fatal Error("Out of space!") ;
-  else
-  {
-      TmpCell->Element = X;
-      TmpCe11->Next = S->Next;
-      S->Next = TmpCell;
-  }
-}
-
        -
        ElementType Top(Stack S)
-{
-  if(!IsEmpty(S))
-      return S->Next->Element;
-  Error("Empty stack") ;
-  return O; /* Return value used to avoid warning*/
-}
-
        -
        void Pop(Stack s)
-{
-  PtrToNode FirstCell;
-  if(IsEmpty(S))
-      Error("Empty stack") ;
-  else
-  {
-      FirstCe11 = S->Next;
-      S->Next = S->Next->Next;
-      free(FirstCe11);
-  }
-}
-
        +

        c + int IsEmpty(Stack S) + { + return S->Next == NULL; + }

        +

        ```c + Stack CreateStack(void) + { + Stack S; + S = malloc(sizeof(struct Node)); + if (S == NULL) + Fatal Error("Out of space!"); + S->Next == NULL; + MakeEmpty(S); + return S; + }

        +

        void MakeEmpty(Stack S) + { + if (S == NULL) + Error("Must use CreateStack first"); + else + while(!IsEmpty(S)) Pop(S); + } + ```

        +

        c + void Push(ElementType X, Stack S) + { + PtrToNode TmpCell; + TmpCell = malloc(sizeof(struct Node)); + if (TmpCell == NULL) + Fatal Error("Out of space!") ; + else + { + TmpCell->Element = X; + TmpCe11->Next = S->Next; + S->Next = TmpCell; + } + }

        +

        c + ElementType Top(Stack S) + { + if(!IsEmpty(S)) + return S->Next->Element; + Error("Empty stack") ; + return O; /* Return value used to avoid warning*/ + }

        +

        c + void Pop(Stack s) + { + PtrToNode FirstCell; + if(IsEmpty(S)) + Error("Empty stack") ; + else + { + FirstCe11 = S->Next; + S->Next = S->Next->Next; + free(FirstCe11); + } + }

        Array Implementation of Stacks

        -
        struct StackRecord {
-    int Capacity;          /* size of stack */
-    int TopOfStack;        /* the top pointer */
-    /* ++ for push, -- for pop, -1 for empty stack */
-    ElementType *Array;    /* array for stack elements */
-}; 
-
        +
        struct StackRecord {
+    int Capacity;          /* size of stack */
+    int TopOfStack;        /* the top pointer */
+    /* ++ for push, -- for pop, -1 for empty stack */
+    ElementType *Array;    /* array for stack elements */
+}; 
+

        • The stack model must be well encapsulated(封装). That is, no part of your code, except for the stack routines, can attempt to access the Array or TopOfStack variable.

          @@ -4067,95 +4074,96 @@

          Array Implementation of Stacks

          Error check must be done before Push or Pop (Top).

        -
        Stack CreateStack(int MaxElements)
-{
-  Stack S;
-  if(MaxElements < MinStackSize)
-  Error("Stack size is too small") ;
-  S = malloc(sizeof(struct StackRecord));
-  if (S == NULL)
-      Fatal Error("Out of space!!!") ;
-
-  S->Array = malloc(sizeof(ElementType) * MaxElements) ;
-  if(S->Array = NULL)
-      Fatal Error("Out of space!!!");
-  S->Capacity = MaxElements;
-  MakeEmpty(S) ;
-  return S;
-}
-
        -
        void DisposeStack(Stack S)
-{
-  if(S != NULL)
-  {
-      free(S->Array);
-      free(S);
-  }
-}
-
        -
        int IsEmpty(Stack S)
-{
-  return S->TopOfStack == EmptyTOS;
-}
-
        -
        void MakeEmpty(Stack S)
-{
-  S->TopOfStack = EmptyTOS;
-}
-
        -
        void Push(ElementType X, Stack S)
-{
-  if (IsFull(S))
-      Error("Full stack");
-  else
-      S->Array[ ++S->TopOfStack ] = X;
-}
-
        -
        ElementType Top(Stack S)
-{
-  if(! IsEmpty(S))
-      return S->Array[ S->TopOfStack ];
-  Error("Empty stack") ;
-  return O; /* Return value used to avoid warning*/
-}
-
        -
        void Pop(Stack S)
-{
-  if(IsEmpty(S))
-      Error("Empty stack") ;
-  else
-      S->TopOfStack--;
-}
-
        -
        ElementType TopAndPop(Stack S)
-{
-  if(!Is Empty(S))
-      return S->Array[ S->TopOfStack-- ];
-  Error("Empty stack");
-  return O; /* Return value used to avoid warnin */
-}
-
        +

        ```c + Stack CreateStack(int MaxElements) + { + Stack S; + if(MaxElements < MinStackSize) + Error("Stack size is too small") ; + S = malloc(sizeof(struct StackRecord)); + if (S == NULL) + Fatal Error("Out of space!!!") ;

        +
        S->Array = malloc(sizeof(ElementType) * MaxElements) ;
+if(S->Array = NULL)
+    Fatal Error("Out of space!!!");
+S->Capacity = MaxElements;
+MakeEmpty(S) ;
+return S;
+
        +

        } + ```

        +

        c + void DisposeStack(Stack S) + { + if(S != NULL) + { + free(S->Array); + free(S); + } + }

        +

        c + int IsEmpty(Stack S) + { + return S->TopOfStack == EmptyTOS; + }

        +

        c + void MakeEmpty(Stack S) + { + S->TopOfStack = EmptyTOS; + }

        +

        c + void Push(ElementType X, Stack S) + { + if (IsFull(S)) + Error("Full stack"); + else + S->Array[ ++S->TopOfStack ] = X; + }

        +

        c + ElementType Top(Stack S) + { + if(! IsEmpty(S)) + return S->Array[ S->TopOfStack ]; + Error("Empty stack") ; + return O; /* Return value used to avoid warning*/ + }

        +

        c + void Pop(Stack S) + { + if(IsEmpty(S)) + Error("Empty stack") ; + else + S->TopOfStack--; + }

        +

        c + ElementType TopAndPop(Stack S) + { + if(!Is Empty(S)) + return S->Array[ S->TopOfStack-- ]; + Error("Empty stack"); + return O; /* Return value used to avoid warnin */ + }

        Application

        1. Balancing Symbols

        检查括号是否平衡

        -
        Algorithm  {
-    Make an empty stack S;
-    while (read in a character c) {
-        if (c is an opening symbol)
-            Push(c, S);
-        else if (c is a closing symbol) {
-            if (S is empty)  { ERROR; exit; }
-            else  {  /* stack is okay */
-                if  (Top(S) doesn’t match c)  { ERROR, exit; }
-                else  Pop(S);
-            }  /* end else-stack is okay */
-        }  /* end else-if-closing symbol */
-    } /* end while-loop */ 
-    if (S is not empty)  ERROR;
-}
-
        +

        pseudocode + Algorithm { + Make an empty stack S; + while (read in a character c) { + if (c is an opening symbol) + Push(c, S); + else if (c is a closing symbol) { + if (S is empty) { ERROR; exit; } + else { /* stack is okay */ + if (Top(S) doesn’t match c) { ERROR, exit; } + else Pop(S); + } /* end else-stack is okay */ + } /* end else-if-closing symbol */ + } /* end while-loop */ + if (S is not empty) ERROR; + }

        1. Postfix Evaluation 后缀表达式

          @@ -4197,14 +4205,14 @@

          2.4 The Queue ADT

        2. Dequeue

      Array Implementation of Queues

      -
      struct QueueRecord {
-    int Capacity ;       /* max size of queue */
-    int Front;           /* the front pointer */
-    int Rear;            /* the rear pointer */
-    int Size;            /* Optional - the current size of queue */
-    ElementType *Array;  /* array for queue elements */
- }; 
-
      +
      struct QueueRecord {
+    int Capacity ;       /* max size of queue */
+    int Front;           /* the front pointer */
+    int Rear;            /* the rear pointer */
+    int Size;            /* Optional - the current size of queue */
+    ElementType *Array;  /* array for queue elements */
+ }; 
+

      Circular Queue :

      3-33-4

        @@ -4261,72 +4269,72 @@

        Tree Traversals (visit eac
        1. Preorder Traversal
        -
        void preorder( tree_ptr tree )
-{ 
-    if( tree )   
-    {
-        visit ( tree );
-        for (each child C of tree )
-            preorder ( C );
-    }
-}
-
        +

        pseudocode + void preorder( tree_ptr tree ) + { + if( tree ) + { + visit ( tree ); + for (each child C of tree ) + preorder ( C ); + } + }

        1. Postorder Traversal
        -
        void postorder( tree_ptr tree )
-{  
- if( tree )   
- {
-        for (each child C of tree )
-         postorder ( C );
-        visit ( tree );
-    }
-}
-
        +

        pseudocode + void postorder( tree_ptr tree ) + { + if( tree ) + { + for (each child C of tree ) + postorder ( C ); + visit ( tree ); + } + }

        1. Levelorder Traversal
        -
        void levelorder( tree_ptr tree )
-{   
- enqueue ( tree );
-    while (queue is not empty) 
-    {
-        visit ( T = dequeue ( ) );
-        for (each child C of T )
-            enqueue ( C );
-    }
-}
-
        +

        pseudocode + void levelorder( tree_ptr tree ) + { + enqueue ( tree ); + while (queue is not empty) + { + visit ( T = dequeue ( ) ); + for (each child C of T ) + enqueue ( C ); + } + }

        1. Inorder Traversal
        -
        void inorder( tree_ptr  tree )
-{  
- if( tree )   
- {
-     inorder ( tree->Left );
-         visit ( tree->Element );
-         inorder ( tree->Right );
-   }
-}
-
        +

        pseudocode + void inorder( tree_ptr tree ) + { + if( tree ) + { + inorder ( tree->Left ); + visit ( tree->Element ); + inorder ( tree->Right ); + } + }

        Iterative Program :

        -
        void iter_inorder( tree_ptr tree )
-{ 
- Stack  S = CreateStack( MAX_SIZE );
- for ( ; ; )  
- {
-     for ( ; tree; tree = tree->Left )
-         Push ( tree, S );
-         tree = Top ( S );  
-         Pop( S );
-         if ( !tree ) break;
-         visit ( tree->Element );
-         tree = tree->Right; 
-    }
-}
-
        +

        pseudocode + void iter_inorder( tree_ptr tree ) + { + Stack S = CreateStack( MAX_SIZE ); + for ( ; ; ) + { + for ( ; tree; tree = tree->Left ) + Push ( tree, S ); + tree = Top ( S ); + Pop( S ); + if ( !tree ) break; + visit ( tree->Element ); + tree = tree->Right; + } + }

        4-3

        Threaded Binary Trees

          @@ -4345,17 +4353,17 @@

          Threaded Binary Trees

        • There must not be any loose threads. Therefore a threaded binary tree must have a head node of which the left child points to the first node.
    -
    typedef struct ThreadedTreeNode *PtrToThreadedNode;
-typedef struct PtrToThreadedNode ThreadedTree;
-typedef struct ThreadedTreeNode 
-{
-    int LeftThread;        /* if it is TRUE, then Left */
-    ThreadedTree Left;     /* is a thread, not a child ptr.*/
-    ElementType Element;
-    int RightThread;       /* if it is TRUE, then Right */
-    ThreadedTree Right;    /* is a thread, not a child ptr.*/
-}
-
    +
    typedef struct ThreadedTreeNode *PtrToThreadedNode;
+typedef struct PtrToThreadedNode ThreadedTree;
+typedef struct ThreadedTreeNode 
+{
+    int LeftThread;        /* if it is TRUE, then Left */
+    ThreadedTree Left;     /* is a thread, not a child ptr.*/
+    ElementType Element;
+    int RightThread;       /* if it is TRUE, then Right */
+    ThreadedTree Right;    /* is a thread, not a child ptr.*/
+}
+
    • 线索化的实质就是将二叉链表中的空指针改为指向前驱或后继的线索。由于前驱和后继信息只有在遍历该二叉树时才能得到，所以，线索化的过程就是在遍历的过程中修改空指针的过程。
    @@ -4396,86 +4404,86 @@

    Implementations

    1. Find
    -
    Position Find( ElementType X, SearchTree T ) 
-{ 
- if ( T == NULL ) 
-     return NULL;  /* not found in an empty tree */
-    if ( X < T->Element )  /* if smaller than root */
-        return Find( X, T->Left );  /* search left subtree */
-    else 
-     if ( X > T->Element )  /* if larger than root */
-         return  Find( X, T->Right );  /* search right subtree */
-        else   /* if X == root */
-         return  T;  /* found */
-} 
-
    +

    pseudocode + Position Find( ElementType X, SearchTree T ) + { + if ( T == NULL ) + return NULL; /* not found in an empty tree */ + if ( X < T->Element ) /* if smaller than root */ + return Find( X, T->Left ); /* search left subtree */ + else + if ( X > T->Element ) /* if larger than root */ + return Find( X, T->Right ); /* search right subtree */ + else /* if X == root */ + return T; /* found */ + }

    • \(T(N)=S(N)=O(d)\) where \(d\) is the depth of X

    Iterative program :

    -
    Position Iter_Find( ElementType X, SearchTree T ) 
-{ 
- while ( T )
- {
-     if ( X == T->Element )  
-         return T;  /* found */
-        if ( X < T->Element )
-            T = T->Left; /*move down along left path */
-        else
-             T = T-> Right; /* move down along right path */
-    }  /* end while-loop */
-    return NULL;   /* not found */
-} 
-
    +

    pseudocode + Position Iter_Find( ElementType X, SearchTree T ) + { + while ( T ) + { + if ( X == T->Element ) + return T; /* found */ + if ( X < T->Element ) + T = T->Left; /*move down along left path */ + else + T = T-> Right; /* move down along right path */ + } /* end while-loop */ + return NULL; /* not found */ + }

    1. FindMin
    -
    Position FindMin( SearchTree T ) 
-{ 
- if ( T == NULL )   
-     return NULL; /* not found in an empty tree */
-    else 
-        if ( T->Left == NULL ) return T;  /* found left most */
-        else return FindMin( T->Left );   /* keep moving to left */
-} 
-
    +

    pseudocode + Position FindMin( SearchTree T ) + { + if ( T == NULL ) + return NULL; /* not found in an empty tree */ + else + if ( T->Left == NULL ) return T; /* found left most */ + else return FindMin( T->Left ); /* keep moving to left */ + }

    1. FindMax
    -
    Position FindMax( SearchTree T ) 
-{ 
- if ( T != NULL ) 
-     while ( T->Right != NULL )   
-         T = T->Right;   /* keep moving to find right most */
-    return T;  /* return NULL or the right most */
-} 
-
    +

    pseudocode + Position FindMax( SearchTree T ) + { + if ( T != NULL ) + while ( T->Right != NULL ) + T = T->Right; /* keep moving to find right most */ + return T; /* return NULL or the right most */ + }

    1. Insert
    -
    SearchTree Insert( ElementType X, SearchTree T ) 
-{ 
-    if ( T == NULL ) /* Create and return a one-node tree */ 
-    { 
-     T = malloc( sizeof( struct TreeNode ) ); 
-     if ( T == NULL ) 
-         FatalError( "Out of space!!!" ); 
-     else 
-     { 
-         T->Element = X; 
-         T->Left = T->Right = NULL; 
-     } 
-    }  /* End creating a one-node tree */
-    else  /* If there is a tree */
-         if ( X < T->Element ) 
-         T->Left = Insert( X, T->Left ); 
-     else 
-         if ( X > T->Element ) 
-             T->Right = Insert( X, T->Right ); 
-     /* Else X is in the tree already; we'll do nothing */ 
-    return  T;   /* Do not forget this line!! */ 
-}
-
    +

    pseudocode + SearchTree Insert( ElementType X, SearchTree T ) + { + if ( T == NULL ) /* Create and return a one-node tree */ + { + T = malloc( sizeof( struct TreeNode ) ); + if ( T == NULL ) + FatalError( "Out of space!!!" ); + else + { + T->Element = X; + T->Left = T->Right = NULL; + } + } /* End creating a one-node tree */ + else /* If there is a tree */ + if ( X < T->Element ) + T->Left = Insert( X, T->Left ); + else + if ( X > T->Element ) + T->Right = Insert( X, T->Right ); + /* Else X is in the tree already; we'll do nothing */ + return T; /* Do not forget this line!! */ + }

    • 内存越界后不会马上报错，在下一次free或malloc时会失败
    • Handle duplicated keys
      • @@ -4491,32 +4499,32 @@

      Implementations

    • Delete a degree 1 node : Replace the node by its single child
    • Delete a degree 2 node : 用左子树最大值结点或右子树最小值结点替换
    -
    SearchTree Delete( ElementType X, SearchTree T ) 
-{    
- Position TmpCell; 
-    if ( T == NULL ) Error( "Element not found" ); 
-    else if ( X < T->Element )  /* Go left */ 
-     T->Left = Delete( X, T->Left ); 
-    else if ( X > T->Element )  /* Go right */ 
-     T->Right = Delete( X, T->Right ); 
- else  /* Found element to be deleted */ 
-     if ( T->Left && T->Right ) {  /* Two children */ 
-     /* Replace with smallest in right subtree */ 
-         TmpCell = FindMin( T->Right ); 
-         T->Element = TmpCell->Element; 
-         T->Right = Delete( T->Element, T->Right );  } /* End if */
-     else 
-     {  /* One or zero child */ 
-         TmpCell = T; 
-         if ( T->Left == NULL ) /* Also handles 0 child */ 
-             T = T->Right; 
-         else if ( T->Right == NULL )  
-             T = T->Left; 
-         free( TmpCell );  
-      }  /* End else 1 or 0 child */
-      return  T; 
-}
-
    +

    pseudocode + SearchTree Delete( ElementType X, SearchTree T ) + { + Position TmpCell; + if ( T == NULL ) Error( "Element not found" ); + else if ( X < T->Element ) /* Go left */ + T->Left = Delete( X, T->Left ); + else if ( X > T->Element ) /* Go right */ + T->Right = Delete( X, T->Right ); + else /* Found element to be deleted */ + if ( T->Left && T->Right ) { /* Two children */ + /* Replace with smallest in right subtree */ + TmpCell = FindMin( T->Right ); + T->Element = TmpCell->Element; + T->Right = Delete( T->Element, T->Right ); } /* End if */ + else + { /* One or zero child */ + TmpCell = T; + if ( T->Left == NULL ) /* Also handles 0 child */ + T = T->Right; + else if ( T->Right == NULL ) + T = T->Left; + free( TmpCell ); + } /* End else 1 or 0 child */ + return T; + }

    • \(T(N)=O(d)\)
    @@ -4627,24 +4635,24 @@

    Structure Property

    None && {2i+1>n}\\ \end{array} \right.\) -
    PriorityQueue Initialize( int MaxElements ) 
-{ 
-    PriorityQueue H; 
-    if ( MaxElements < MinPQSize ) 
-        return Error( "Priority queue size is too small" ); 
-    H = malloc(sizeof( struct HeapStruct )); 
-    if ( H == NULL ) 
-        return FatalError( "Out of space!!!" ); 
-    /* Allocate the array plus one extra for sentinel */ 
-    H->Elements = malloc(( MaxElements + 1 ) * sizeof( ElementType )); 
-    if ( H->Elements == NULL ) 
-        return FatalError( "Out of space!!!" ); 
-    H->Capacity = MaxElements; 
-    H->Size = 0; 
-    H->Elements[0] = MinData;  /* set the sentinel */
-    return H; 
-}
-
    +
    PriorityQueue Initialize( int MaxElements ) 
+{ 
+    PriorityQueue H; 
+    if ( MaxElements < MinPQSize ) 
+        return Error( "Priority queue size is too small" ); 
+    H = malloc(sizeof( struct HeapStruct )); 
+    if ( H == NULL ) 
+        return FatalError( "Out of space!!!" ); 
+    /* Allocate the array plus one extra for sentinel */ 
+    H->Elements = malloc(( MaxElements + 1 ) * sizeof( ElementType )); 
+    if ( H->Elements == NULL ) 
+        return FatalError( "Out of space!!!" ); 
+    H->Capacity = MaxElements; 
+    H->Size = 0; 
+    H->Elements[0] = MinData;  /* set the sentinel */
+    return H; 
+}
+

    Heap Order Property

    [Definition] A min tree is a tree in which the key value in each node is no larger than the key values in its children (if any). A min heap is a complete binary tree that is also a min tree.

      @@ -4654,51 +4662,51 @@

      Basic Heap Operations

      1. Insertion
      -
      /*H->Element[ 0 ] is a sentinel that is no larger than the minimum element in the heap.*/ 
-void Insert( ElementType X, PriorityQueue H ) 
-{ 
- int i; 
-    if ( IsFull( H )) 
-    { 
-     Error( "Priority queue is full" ); 
-     return; 
-    } 
-    for ( i = ++H->Size; H->Elements[ i/2 ] > X; i /= 2 ) 
-     H->Elements[ i ] = H->Elements[ i/2 ]; /*Percolate up, faster than swap*/
-    H->Elements[ i ] = X; 
-}
-
      +

      c + /*H->Element[ 0 ] is a sentinel that is no larger than the minimum element in the heap.*/ + void Insert( ElementType X, PriorityQueue H ) + { + int i; + if ( IsFull( H )) + { + Error( "Priority queue is full" ); + return; + } + for ( i = ++H->Size; H->Elements[ i/2 ] > X; i /= 2 ) + H->Elements[ i ] = H->Elements[ i/2 ]; /*Percolate up, faster than swap*/ + H->Elements[ i ] = X; + }

      $$ T(N)=O(\log N) $$

      1. DeleteMin
      -
      ElementType DeleteMin( PriorityQueue H ) 
-{ 
-    int i, Child; 
-    ElementType MinElement, LastElement; 
-    if ( IsEmpty( H ) ) 
-    { 
-        Error( "Priority queue is empty" ); 
-        return H->Elements[ 0 ];   
-    } 
-    MinElement = H->Elements[ 1 ];  /*Save the min element*/
-    LastElement = H->Elements[ H->Size-- ];  /*Take last and reset size*/
-    for ( i = 1; i * 2 <= H->Size; i = Child )  /*Find smaller child*/ 
-    {
-        Child = i * 2; 
-        if (Child != H->Size && H->Elements[Child+1] < H->Elements[Child]) 
-             Child++;     
-        if ( LastElement > H->Elements[ Child ] )   /*Percolate one level*/ 
-             H->Elements[ i ] = H->Elements[ Child ]; 
-        else     
-         break;   /*Find the proper position*/
-    } 
-    H->Elements[ i ] = LastElement; 
-    return MinElement; 
-}
-
      +

      c + ElementType DeleteMin( PriorityQueue H ) + { + int i, Child; + ElementType MinElement, LastElement; + if ( IsEmpty( H ) ) + { + Error( "Priority queue is empty" ); + return H->Elements[ 0 ]; + } + MinElement = H->Elements[ 1 ]; /*Save the min element*/ + LastElement = H->Elements[ H->Size-- ]; /*Take last and reset size*/ + for ( i = 1; i * 2 <= H->Size; i = Child ) /*Find smaller child*/ + { + Child = i * 2; + if (Child != H->Size && H->Elements[Child+1] < H->Elements[Child]) + Child++; + if ( LastElement > H->Elements[ Child ] ) /*Percolate one level*/ + H->Elements[ i ] = H->Elements[ Child ]; + else + break; /*Find the proper position*/ + } + H->Elements[ i ] = LastElement; + return MinElement; + }

      \[ T(N)=O(\log N) \]
      @@ -4721,9 +4729,9 @@

      Other Heap Operations

    将N 个关键字以任意顺序放入树中，保持结构特性，再执行下滤

    -
    for (i = N/2; i > 0; i--)
- PercolateDown(i);
-
    +

    c + for (i = N/2; i > 0; i--) + PercolateDown(i);

    $$ T(N)=O(N) $$

    @@ -4734,31 +4742,31 @@

    4.4 Applications of Priority Queues<

    Heap Sort

    查找一个序列中第k小的元素

    The function is to find the K-th smallest element in a list A of N elements. The function BuildMaxHeap(H, K) is to arrange elements H[1] ... H[K] into a max-heap.

    -
    ElementType FindKthSmallest ( int A[], int N, int K )
-{   /* it is assumed that K<=N */
-    ElementType *H;
-    int i, next, child;
-
-    H = (ElementType*)malloc((K+1)*sizeof(ElementType));
-    for ( i = 1; i <= K; i++ ) H[i] = A[i-1];
-    BuildMaxHeap(H, K);
-
-    for ( next = K; next < N; next++ ) {
-        H[0] = A[next];
-        if ( H[0] < H[1] ) {
-            for ( i = 1; i*2 <= K; i = child ) {
-                child = i*2;
-                if ( child != K && H[child+1] > H[child] ) child++;
-                if ( H[0] < H[child] )
-                    H[i] = H[child];
-                else break;
-            }
-            H[i] = H[0];
-        }
-    }
-    return H[1];
-}
-
    +
    ElementType FindKthSmallest ( int A[], int N, int K )
+{   /* it is assumed that K<=N */
+    ElementType *H;
+    int i, next, child;
+
+    H = (ElementType*)malloc((K+1)*sizeof(ElementType));
+    for ( i = 1; i <= K; i++ ) H[i] = A[i-1];
+    BuildMaxHeap(H, K);
+
+    for ( next = K; next < N; next++ ) {
+        H[0] = A[next];
+        if ( H[0] < H[1] ) {
+            for ( i = 1; i*2 <= K; i = child ) {
+                child = i*2;
+                if ( child != K && H[child+1] > H[child] ) child++;
+                if ( H[0] < H[child] )
+                    H[i] = H[child];
+                else break;
+            }
+            H[i] = H[0];
+        }
+    }
+    return H[1];
+}
+

    4.5 \(d\)-Heaps — All nodes have \(d\) children

    Note :

    @@ -4783,23 +4791,23 @@

    5.2 The Dynamic Equivalence Problem<
    • Given an equivalence relation ~, decide for any a and b if a ~ b
    -
    Algorithm: (Union/Find)
-{
-  /* step 1: read the relations in */
-    Initialize N disjoint sets;
-    while ( read in a ~ b )
-    {
-      if ( !(Find(a) == Find(b)) )  /*Dynamic(on-line)*/
-          Union the two sets;
-    } /* end-while */
-    /* step 2: decide if a ~ b */
-    while ( read in a and b )
-        if ( Find(a) == Find(b) )
-          output( true );
-        else   
-          output( false );
-}
-
    +

    pseudocode + Algorithm: (Union/Find) + { + /* step 1: read the relations in */ + Initialize N disjoint sets; + while ( read in a ~ b ) + { + if ( !(Find(a) == Find(b)) ) /*Dynamic(on-line)*/ + Union the two sets; + } /* end-while */ + /* step 2: decide if a ~ b */ + while ( read in a and b ) + if ( Find(a) == Find(b) ) + output( true ); + else + output( false ); + }

    • Elements of the sets : \(1,2,3,\cdots,N\)

      @@ -4842,11 +4850,11 @@

      Union( \(i, j\) )

    • 数组初始化全部为0
    -
    void SetUnion(DisjSet S, SetType Rt1, SetType Rt2)
-{
-  S[Rt2] = Rt1;
-}
-
    +

    c + void SetUnion(DisjSet S, SetType Rt1, SetType Rt2) + { + S[Rt2] = Rt1; + }

    Find( \(i\) )

    • Implementation 1 :
      • @@ -4855,18 +4863,18 @@

      Find( \(i\) )

      • Implementation 2 :
      -
      SetType Find(ElementType X, DisjSet S)
-{
-  for ( ; S[X]>0; X=S[X]);
-  return X;
-}
-
      +

      c + SetType Find(ElementType X, DisjSet S) + { + for ( ; S[X]>0; X=S[X]); + return X; + }

      Analysis

      • Union and find are always paired. Thus we consider the performance of a sequence of union-find operations.

      7-3

      -
      Algorithm using union-find operations:
+
      Algorithm using union-find operations:
 {  
     Initialize Si = { i }  for  i = 1, ..., 12 ;
     for ( k = 1; k <= 9; k++ )  /* for each pair i R j */
@@ -4875,7 +4883,7 @@ 
      Analysis
      
             SetUnion( Find( i ), Find( j ) );
     }
 }
-
      +
      • Worst case : \(T(N)=\Theta(N^2)\)
      @@ -4897,64 +4905,64 @@

      Union-by-Size

      • Time complexity of \(N\) Union and \(M\) Find operations is now \(O(N+M\log_2N)\).
      -
      /* Assumes Rootl and Root2 are roots*/
-void SetUnion(DisjSet S, SetType Root1, SetType Root2)
-{
-    if (S[Root1] <= S[Root2])
-    {
-        S[Root1] += S[Root2];
-        S[Root2] = Root1;
-    }
-    else
-    {
-        S[Root2] += S[Root1];
-        S[Root1] = Root2;
-    }
-}
-
      +
      /* Assumes Rootl and Root2 are roots*/
+void SetUnion(DisjSet S, SetType Root1, SetType Root2)
+{
+    if (S[Root1] <= S[Root2])
+    {
+        S[Root1] += S[Root2];
+        S[Root2] = Root1;
+    }
+    else
+    {
+        S[Root2] += S[Root1];
+        S[Root1] = Root2;
+    }
+}
+

      Union-by-Height

      • Always change the shallow tree
      • 保证所有的树的深度最多是\(O(logN)\)
      -
      /* Assumes Rootl and Root2 are roots*/
-void SetUnion(DisjSet S, SetType Root1, SetType Root2)
-{
-    if ( S[Root2] < S[Root1])  /*Root2 is deeper set*/
-        S[Root1] = Root2;      /*Make Root2 new root*/
-    else
-    {
-        if (S[Root1] == S[Root2])  /*Same height*/
-            S[Root1]--;
-        S[Root2] = Root1;
-    }
-}
-
      +
      /* Assumes Rootl and Root2 are roots*/
+void SetUnion(DisjSet S, SetType Root1, SetType Root2)
+{
+    if ( S[Root2] < S[Root1])  /*Root2 is deeper set*/
+        S[Root1] = Root2;      /*Make Root2 new root*/
+    else
+    {
+        if (S[Root1] == S[Root2])  /*Same height*/
+            S[Root1]--;
+        S[Root2] = Root1;
+    }
+}
+

      5.5 Path Compression

      • 从X到Root的路径上的每一个结点都使它的父结点变成Root
      -
      SetType Find( ElementType X, DisjSet S )
-{
-    if ( S[ X ] <= 0 )    
-        return X;
-    else 
-        return S[ X ] = Find( S[ X ], S );
-}
-
      -
      SetType Find( ElementType X, DisjSet S )
-{   
-    ElementType root, trail, lead;
-    for ( root = X; S[ root ] > 0; root = S[ root ] );  /* find the root */
-    for ( trail = X; trail != root; trail = lead )
-    {
-        lead = S[ trail ];   
-        S[ trail ] = root;   
-    }  /* collapsing */
-    return root;
-}
-
      +
      SetType Find( ElementType X, DisjSet S )
+{
+    if ( S[ X ] <= 0 )    
+        return X;
+    else 
+        return S[ X ] = Find( S[ X ], S );
+}
+
      +
      SetType Find( ElementType X, DisjSet S )
+{   
+    ElementType root, trail, lead;
+    for ( root = X; S[ root ] > 0; root = S[ root ] );  /* find the root */
+    for ( trail = X; trail != root; trail = lead )
+    {
+        lead = S[ trail ];   
+        S[ trail ] = root;   
+    }  /* collapsing */
+    return root;
+}
+
      • Note : Not compatible with union-by-height since it changes the heights. Just take “height” as an estimated rank.
      @@ -5145,50 +5153,50 @@ 

      /*Test an AOV for feasibility, and generate a topological order if possible*/
-void Topsort( Graph G )
-{   
-    int Counter;
-    Vertex V, W;
-    for ( Counter = 0; Counter < NumVertex; Counter++ ) 
-    {
-        V = FindNewVertexOfDegreeZero( );
-        if ( V == NotAVertex ) 
-        {
-            Error ( “Graph has a cycle” );   
-            break;  
-        }
-        TopNum[ V ] = Counter; /* or output V */
-        for ( each W adjacent to V )
-            Indegree[ W ]––;
-    }
-}
-
      +
      /*Test an AOV for feasibility, and generate a topological order if possible*/
+void Topsort( Graph G )
+{   
+    int Counter;
+    Vertex V, W;
+    for ( Counter = 0; Counter < NumVertex; Counter++ ) 
+    {
+        V = FindNewVertexOfDegreeZero( );
+        if ( V == NotAVertex ) 
+        {
+            Error ( “Graph has a cycle” );   
+            break;  
+        }
+        TopNum[ V ] = Counter; /* or output V */
+        for ( each W adjacent to V )
+            Indegree[ W ]––;
+    }
+}
+
      \[ T=O(|V|^2) \]
      -
      /*Improvment:Keep all the unassigned vertices of degree 0 in a special box (queue or stack)*/
-void Topsort( Graph G )
-{   
-    Queue Q;
-    int Counter = 0;
-    Vertex V, W;
-    Q = CreateQueue( NumVertex );  
-    MakeEmpty( Q );
-    for ( each vertex V )
-        if ( Indegree[ V ] == 0 ) Enqueue( V, Q );
-    while ( !IsEmpty( Q ) ) 
-    {
-        V = Dequeue( Q );
-        TopNum[ V ] = ++Counter; /* assign next */
-        for ( each W adjacent to V )
-            if (––Indegree[ W ] == 0 ) Enqueue( W, Q );
-    }  /* end-while */
-    if ( Counter != NumVertex )
-    Error( “Graph has a cycle” );
-    DisposeQueue( Q ); /* free memory */
-}
-
      +
      /*Improvment:Keep all the unassigned vertices of degree 0 in a special box (queue or stack)*/
+void Topsort( Graph G )
+{   
+    Queue Q;
+    int Counter = 0;
+    Vertex V, W;
+    Q = CreateQueue( NumVertex );  
+    MakeEmpty( Q );
+    for ( each vertex V )
+        if ( Indegree[ V ] == 0 ) Enqueue( V, Q );
+    while ( !IsEmpty( Q ) ) 
+    {
+        V = Dequeue( Q );
+        TopNum[ V ] = ++Counter; /* assign next */
+        for ( each W adjacent to V )
+            if (––Indegree[ W ] == 0 ) Enqueue( W, Q );
+    }  /* end-while */
+    if ( Counter != NumVertex )
+    Error( “Graph has a cycle” );
+    DisposeQueue( Q ); /* free memory */
+}
+
      \[ T=O(|V|+|E|) \]
      @@ -5211,55 +5219,55 @@

      Unweighted Shortest Path
    • Table[ i ].Known ::= 1 if \(v_i\) is checked; or 0 if not
    • Table[ i ].Path ::= for tracking the path / initialized to be 0 /
    -
    void Unweighted( Table T )
-{   
-    int CurrDist;
-    Vertex V, W;
-    for ( CurrDist = 0; CurrDist < NumVertex; CurrDist++ ) 
-    {
-        for ( each vertex V )
-            if ( !T[ V ].Known && T[ V ].Dist == CurrDist ) 
-            {
-                T[ V ].Known = true;
-                for ( each W adjacent to V )
-                    if ( T[ W ].Dist == Infinity ) 
-                    {
-                        T[ W ].Dist = CurrDist + 1;
-                        T[ W ].Path = V;
-                    } /* end-if Dist == Infinity */
-            } /* end-if !Known && Dist == CurrDist */
-    }  /* end-for CurrDist */
-}
-
    +
    void Unweighted( Table T )
+{   
+    int CurrDist;
+    Vertex V, W;
+    for ( CurrDist = 0; CurrDist < NumVertex; CurrDist++ ) 
+    {
+        for ( each vertex V )
+            if ( !T[ V ].Known && T[ V ].Dist == CurrDist ) 
+            {
+                T[ V ].Known = true;
+                for ( each W adjacent to V )
+                    if ( T[ W ].Dist == Infinity ) 
+                    {
+                        T[ W ].Dist = CurrDist + 1;
+                        T[ W ].Path = V;
+                    } /* end-if Dist == Infinity */
+            } /* end-if !Known && Dist == CurrDist */
+    }  /* end-for CurrDist */
+}
+

    The worst case :

    9-1 $$ T(N)=O(|V|^2) $$ Improvement :

    -
    void Unweighted( Table T )
-{   
-    /* T is initialized with the source vertex S given */
-    Queue Q;
-    Vertex V, W;
-    Q = CreateQueue( NumVertex );
-    MakeEmpty( Q );
-    Enqueue( S, Q ); /* Enqueue the source vertex */
-    while ( !IsEmpty( Q ) ) 
-    {
-        V = Dequeue( Q );
-        T[ V ].Known = true; /* not really necessary */
-        for ( each W adjacent to V )
-            if ( T[ W ].Dist == Infinity ) 
-            {
-                T[ W ].Dist = T[ V ].Dist + 1;
-                T[ W ].Path = V;
-                Enqueue( W, Q );
-            } /* end-if Dist == Infinity */
-    } /* end-while */
-    DisposeQueue( Q ); /* free memory */
-}
-
    +
    void Unweighted( Table T )
+{   
+    /* T is initialized with the source vertex S given */
+    Queue Q;
+    Vertex V, W;
+    Q = CreateQueue( NumVertex );
+    MakeEmpty( Q );
+    Enqueue( S, Q ); /* Enqueue the source vertex */
+    while ( !IsEmpty( Q ) ) 
+    {
+        V = Dequeue( Q );
+        T[ V ].Known = true; /* not really necessary */
+        for ( each W adjacent to V )
+            if ( T[ W ].Dist == Infinity ) 
+            {
+                T[ W ].Dist = T[ V ].Dist + 1;
+                T[ W ].Path = V;
+                Enqueue( W, Q );
+            } /* end-if Dist == Infinity */
+    } /* end-while */
+    DisposeQueue( Q ); /* free memory */
+}
+
    \[ T=O(|V|+|E|) \]
    @@ -5272,60 +5280,60 @@
    Dijkstra’s Algorithm
  • Greedy Method : \(u\) is chosen so that distance[ u ] = min{ \(w \notin S\) | distance[ w ] } (If \(u\) is not unique, then we may select any of them)
  • if distance[\(u_1\)] < distance[\(u_2\)] and add \(u_1\) into \(S\), then distance [ \(u_2\) ] may change. If so, a shorter path from \(s\) to \(u_2\) must go through \(u_1\) and distance [ \(u_2\) ] = distance [ \(u_1\) ] + length(< \(u_1\), \(u_2\)>).
    • -
    typedef int Vertex;
-struct TableEntry
-{
-    List Header; /*Adjacency list*/
-    int Known;
-    DistType Dist;
-    Vertex Path;
-};
-/*Vertices are numbered from 0*/
-#define NotAVertex (-1)
-typedef struct TableEntry Table[ NumVertex ];
-
    -
    void InitTable(Vertex Start, Graph G, Table T)
-{ 
-    int i;
-    ReadGraph(G, T); /* Read graph somehow */
-    for(i = 0; i < NumVertex; i++)
-    {
-        T[ i ].Known = False;
-        T[ i ].Dist = Infinity;
-        T[ i ].Path = NotAVertex;
-    }
-    T[ Start ].dist = O;
-}
-
    -
    /*Print shortest path to V after Dijkstra has run*/
-/*Assume that the path exists*/
-void PrintPath(Vertex V, Table T)
-{
-    if (T[ V ].Path != NotAVertex)
-    {
-        PrintPath(T[ V ].Path, T);
-        printf(" to") ;
-    }
-    printf("%v", V) ; /* %v is pseudocode * /
-
    -
    void Dijkstra( Table T )
-{ 
-    Vertex V, W;
-    for ( ; ; ) 
-    {
-        V = smallest unknown distance vertex;
-        if ( V == NotAVertex ) break; 
-        T[ V ].Known = true;
-        for ( each W adjacent to V )
-            if ( !T[ W ].Known ) 
-                if ( T[ V ].Dist + Cvw < T[ W ].Dist ) 
-                {
-                    Decrease( T[ W ].Dist to T[ V ].Dist + Cvw );
-                    T[ W ].Path = V;
-                } /* end-if update W */
-    } /* end-for( ; ; ) */
-}
-
    +
    typedef int Vertex;
+struct TableEntry
+{
+    List Header; /*Adjacency list*/
+    int Known;
+    DistType Dist;
+    Vertex Path;
+};
+/*Vertices are numbered from 0*/
+#define NotAVertex (-1)
+typedef struct TableEntry Table[ NumVertex ];
+
    +
    void InitTable(Vertex Start, Graph G, Table T)
+{ 
+    int i;
+    ReadGraph(G, T); /* Read graph somehow */
+    for(i = 0; i < NumVertex; i++)
+    {
+        T[ i ].Known = False;
+        T[ i ].Dist = Infinity;
+        T[ i ].Path = NotAVertex;
+    }
+    T[ Start ].dist = O;
+}
+
    +
    /*Print shortest path to V after Dijkstra has run*/
+/*Assume that the path exists*/
+void PrintPath(Vertex V, Table T)
+{
+    if (T[ V ].Path != NotAVertex)
+    {
+        PrintPath(T[ V ].Path, T);
+        printf(" to") ;
+    }
+    printf("%v", V) ; /* %v is pseudocode * /
+
    +
    void Dijkstra( Table T )
+{ 
+    Vertex V, W;
+    for ( ; ; ) 
+    {
+        V = smallest unknown distance vertex;
+        if ( V == NotAVertex ) break; 
+        T[ V ].Known = true;
+        for ( each W adjacent to V )
+            if ( !T[ W ].Known ) 
+                if ( T[ V ].Dist + Cvw < T[ W ].Dist ) 
+                {
+                    Decrease( T[ W ].Dist to T[ V ].Dist + Cvw );
+                    T[ W ].Path = V;
+                } /* end-if update W */
+    } /* end-for( ; ; ) */
+}
+
    Implementation 1
    • Simply scan the table to find the smallest unknown distance vertex.——\(O(|V|)\)
      • @@ -5364,28 +5372,28 @@
      Improvements
    • Fibonacci heap
    Graphs with Negative Edge Costs
    -
    void WeightedNegative( Table T )
-{
-    Queue Q;
-    Vertex V, W;
-    Q = CreateQueue (NumVertex );  
-    MakeEmpty( Q );
-    Enqueue( S, Q ); /*Enqueue the source vertex*/
-    while ( !IsEmpty( Q ) ) 
-    {
-        V = Dequeue( Q );
-        for ( each W adjacent to V )
-        if ( T[ V ].Dist + Cvw < T[ W ].Dist ) 
-        {
-            T[ W ].Dist = T[ V ].Dist + Cvw;
-            T[ W ].Path = V;
-            if ( W is not already in Q )
-                Enqueue( W, Q );
-        } /*end-if update*/
-    } /*end-while */
-    DisposeQueue( Q ); /*free memory*/
-}
-
    +
    void WeightedNegative( Table T )
+{
+    Queue Q;
+    Vertex V, W;
+    Q = CreateQueue (NumVertex );  
+    MakeEmpty( Q );
+    Enqueue( S, Q ); /*Enqueue the source vertex*/
+    while ( !IsEmpty( Q ) ) 
+    {
+        V = Dequeue( Q );
+        for ( each W adjacent to V )
+        if ( T[ V ].Dist + Cvw < T[ W ].Dist ) 
+        {
+            T[ W ].Dist = T[ V ].Dist + Cvw;
+            T[ W ].Path = V;
+            if ( W is not already in Q )
+                Enqueue( W, Q );
+        } /*end-if update*/
+    } /*end-while */
+    DisposeQueue( Q ); /*free memory*/
+}
+

    Note : Negative-cost cycle will cause indefinite loop

    @@ -5526,50 +5534,51 @@
    Greedy Method

  • 连续地按照最小的权选择边,，并且当所选的边不产生环时就把它作为取定的边

    -
    void Kruskal( Graph G )
-{   
-   T = { };
-    while ( T contains less than |V|-1 edges && E is not empty ) 
+
    pseudocode
+ void Kruskal( Graph G )
+ {   
+    T = { };
+     while ( T contains less than |V|-1 edges && E is not empty ) 
+     {
+         choose a least cost edge (v, w) from E;  /*DeleteMin*/
+         delete (v, w) from E;
+         if ( (v, w) does not create a cycle in T )     
+            add (v, w) to T;  /*Union/Find*/
+         else     
+            discard (v, w);
+     }
+     if ( T contains fewer than |V|-1 edges )
+        Error( “No spanning tree” );
+ }
    
+
    ```c
+ void Kruskal(Graph G)
+ {
+    int EdgesAccepted;
+    DisjSet S;
+    PriorityQueue H;
+    Vertex U, V;
+    SetType Uset, Vset;
+    Edge E;
    
+
    Initialize(S);
+ReadGraphIntoHeapArray(G, H);
+BuildHeap(H);
+
+EdgesAccepted = 0;
+while(EdgesAccepted < NumVertex-1)
+{
+    E = DeleteMin(H); /*E = (U,V)*/
+    Uset = Find(U, S);
+    Vset = Find(V, S);
+    if(Uset != Vset)
     {
-        choose a least cost edge (v, w) from E;  /*DeleteMin*/
-        delete (v, w) from E;
-        if ( (v, w) does not create a cycle in T )     
-           add (v, w) to T;  /*Union/Find*/
-        else     
-           discard (v, w);
+        /*Accept the edge*/
+        EdgesAccepted++;
+        SetUnion(S, USet, VSet);
     }
-    if ( T contains fewer than |V|-1 edges )
-       Error( “No spanning tree” );
 }
-
    -
    void Kruskal(Graph G)
-{
-   int EdgesAccepted;
-   DisjSet S;
-   PriorityQueue H;
-   Vertex U, V;
-   SetType Uset, Vset;
-   Edge E;
-
-   Initialize(S);
-   ReadGraphIntoHeapArray(G, H);
-   BuildHeap(H);
-
-   EdgesAccepted = 0;
-   while(EdgesAccepted < NumVertex-1)
-   {
-       E = DeleteMin(H); /*E = (U,V)*/
-       Uset = Find(U, S);
-       Vset = Find(V, S);
-       if(Uset != Vset)
-       {
-           /*Accept the edge*/
-           EdgesAccepted++;
-           SetUnion(S, USet, VSet);
-       }
-   }
-}
-
    +     +

    } + ```

    $$ T=O(|E|\log|E|) $$

    @@ -5578,25 +5587,25 @@
    Greedy Method
    • -
    /*a generalization of preorder traversal*/
-void DFS(Vertex V)
-{   
-    visited[ V ] = true;  /*mark this vertex to avoid cycles*/
-    for ( each W adjacent to V )
-        if ( !visited[ W ] ) DFS( W );
-} /*T = O(|E|+|V|) as long as adjacency lists are used*/
-
    +
    /*a generalization of preorder traversal*/
+void DFS(Vertex V)
+{   
+    visited[ V ] = true;  /*mark this vertex to avoid cycles*/
+    for ( each W adjacent to V )
+        if ( !visited[ W ] ) DFS( W );
+} /*T = O(|E|+|V|) as long as adjacency lists are used*/
+

    Undirected Graphs

    -
    void ListComponents(Graph G)
-{   
-    for ( each V in G ) 
-        if ( !visited[ V ] ) 
-        {
-            DFS( V );
-            printf("\n");
-        }
-}
-
    +
    void ListComponents(Graph G)
+{   
+    for ( each V in G ) 
+        if ( !visited[ V ] ) 
+        {
+            DFS( V );
+            printf("\n");
+        }
+}
+

    Biconnectivity

    • \(v\) is an articulation point if \(G'=DeleteVertex(G, v)\) has at least 2 connected components.
      • @@ -5638,7 +5647,7 @@

      Biconnectivity

      Low(u)=\min{Num(u),\min{Low(w)|w\,is\,a\,child\,of\,u},\min{Num(w)|(u,w)\,is\,a\,back\,edge}} $$
    -
    /*Assign Num and compute Parents*/
+
    /*Assign Num and compute Parents*/
 void AssignNum(Vertex V)
 {
     Vertex W;
@@ -5651,8 +5660,8 @@ 
    Biconnectivity
    
             AssignNum(W);
         }
 }
-
    -
    /*Assign Low; also check for articulation points*/
+
    +
    /*Assign Low; also check for articulation points*/
 void AssignLow(Vertex V)
 {
     Vertex W;
@@ -5671,8 +5680,8 @@ 
    Biconnectivity
    
                 Low[V] ＝ Min(Low[V], Num[W]); /*Rule 2*/
     }
 }
-
    -
    void FindArt(Vertex V)
+
    +
    void FindArt(Vertex V)
 {
     Vertex W;
     Visited[V] = True;
@@ -5692,7 +5701,7 @@ 
    Biconnectivity
    
                 Low[V] = Min(Low[V], Num[W]); /*Rule 2*/
     }
 }
-
    +

    Euler Circuits

    [Proposition] An Euler circuit is possible only if the graph is connected and each vertex has an even degree.
    [Proposition] An Euler tour is possible if there are exactly two vertices having odd degree. One must start at one of the odd-degree vertices.
    @@ -5707,8 +5716,8 @@
    7 Sorting

    7.1 Preliminaries

    -
    void X_Sort (ElementType A[], int N)
-
    +
    void X_Sort (ElementType A[], int N)
+
    • N must be a legal integer.
    • Assume integer array for the sake of simplicity.
      • @@ -5717,21 +5726,21 @@

      7.1 Preliminaries

    7.2 Insertion Sort

    -
    void Insertion(ElementType A[], int N)
-{ 
-    int j, P; 
-    ElementType Tmp; 
-
-    for ( P = 1; P < N; P++ ) 
-    { 
-        Tmp = A[ P ];  /*the next coming card*/
-        for ( j = P; j > 0 && A[ j - 1 ] > Tmp; j-- ) 
-            A[ j ] = A[ j - 1 ]; 
-            /*shift sorted cards to provide a position for the new coming card*/
-        A[ j ] = Tmp;  /*place the new card at the proper position*/
-    }/*end for-P-loop*/
-}
-
    +
    void Insertion(ElementType A[], int N)
+{ 
+    int j, P; 
+    ElementType Tmp; 
+
+    for ( P = 1; P < N; P++ ) 
+    { 
+        Tmp = A[ P ];  /*the next coming card*/
+        for ( j = P; j > 0 && A[ j - 1 ] > Tmp; j-- ) 
+            A[ j ] = A[ j - 1 ]; 
+            /*shift sorted cards to provide a position for the new coming card*/
+        A[ j ] = Tmp;  /*place the new card at the proper position*/
+    }/*end for-P-loop*/
+}
+

    • The worst case : Input A[ ] is in reverse order @@ -5778,23 +5787,23 @@

      Shell’s Increment Sequence

      \[ h_t=\lfloor N/2\rfloor,h_k=\lfloor h_{k+1}/2\rfloor \]
      -
      void Shellsort( ElementType A[ ], int N ) 
-{ 
-    int i, j, Increment; 
-    ElementType Tmp; 
-    for ( Increment = N / 2; Increment > 0; Increment /= 2 )  /*h sequence */
-        for ( i = Increment; i < N; i++ ) 
-        { /* insertion sort */
-            Tmp = A[ i ]; 
-            for ( j = i; j >= Increment; j -= Increment ) 
-                if( Tmp < A[ j-Increment ] )
-                    A[ j ] = A[ j-Increment ]; 
-                else 
-                    break; 
-            A[ j ] = Tmp;
-        } /* end for-I and for-Increment loops */
-}
-
      +
      void Shellsort( ElementType A[ ], int N ) 
+{ 
+    int i, j, Increment; 
+    ElementType Tmp; 
+    for ( Increment = N / 2; Increment > 0; Increment /= 2 )  /*h sequence */
+        for ( i = Increment; i < N; i++ ) 
+        { /* insertion sort */
+            Tmp = A[ i ]; 
+            for ( j = i; j >= Increment; j -= Increment ) 
+                if( Tmp < A[ j-Increment ] )
+                    A[ j ] = A[ j-Increment ]; 
+                else 
+                    break; 
+            A[ j ] = Tmp;
+        } /* end for-I and for-Increment loops */
+}
+
      • [Theorem] The worst-case running time of Shellsort, using Shell’s increments, is \(\Theta( N^2 )\).
      @@ -5819,15 +5828,15 @@

      Conclusion

      7.5 Heapsort

      Algorithm1

      -
      void Heapsort( int N ) 
-{
-    BuildHeap( H );
-    for ( i = 0; i < N; i++ ) 
-        TmpH[ i ] = DeleteMin( H );
-    for ( i = 0; i < N; i++ ) 
-        H[ i ] = TmpH[ i ];
-}
-
      +
      void Heapsort( int N ) 
+{
+    BuildHeap( H );
+    for ( i = 0; i < N; i++ ) 
+        TmpH[ i ] = DeleteMin( H );
+    for ( i = 0; i < N; i++ ) 
+        H[ i ] = TmpH[ i ];
+}
+
      \[ T(N)=O(N\log N) \]
      @@ -5835,18 +5844,18 @@

      Algorithm1

    • The space requirement is doubled.

    Algorithm2

    -
    void Heapsort( ElementType A[ ], int N ) 
-{
-    int i; 
-    for ( i = N / 2; i >= 0; i-- ) /*BuildHeap*/ 
-        PercDown( A, i, N );
-    for ( i = N - 1; i > 0; i-- ) 
-    { 
-        Swap( &A[ 0 ], &A[ i ] ); /*DeleteMax*/ 
-        PercDown( A, 0, i ); 
-    } 
-}
-
    +
    void Heapsort( ElementType A[ ], int N ) 
+{
+    int i; 
+    for ( i = N / 2; i >= 0; i-- ) /*BuildHeap*/ 
+        PercDown( A, i, N );
+    for ( i = N - 1; i > 0; i-- ) 
+    { 
+        Swap( &A[ 0 ], &A[ i ] ); /*DeleteMax*/ 
+        PercDown( A, 0, i ); 
+    } 
+}
+
    • [Theorem] The average number of comparisons used to heapsort a random permutation of N distinct items is \(2N\log N-O(N\log\log N)\).
    @@ -5855,54 +5864,54 @@

    Algorithm2

    7.6 Mergesort

    -
    void MSort( ElementType A[ ], ElementType TmpArray[ ], int Left, int Right ) 
-{   
-    int Center; 
-    if ( Left < Right ) 
-    {  /*if there are elements to be sort*/
-        Center = (Left+Right)/2; 
-        MSort(A, TmpArray, Left, Center);   /*T(N/2)*/
-        MSort(A, TmpArray, Center+1, Right);    /*T(N/2)*/
-        Merge(A, TmpArray, Left, Center+1, Right);  /*O(N)*/
-    } 
-} 
-
-void Mergesort( ElementType A[ ], int N ) 
-{   
-    ElementType *TmpArray;  /*need O(N) extra space*/
-    TmpArray = malloc(N*sizeof(ElementType)); 
-    if (TmpArray != NULL) 
-    { 
-        MSort(A, TmpArray, 0, N-1); 
-        free(TmpArray); 
-    } 
-    else FatalError("No space for tmp array!!!"); 
-}
-
    +
    void MSort( ElementType A[ ], ElementType TmpArray[ ], int Left, int Right ) 
+{   
+    int Center; 
+    if ( Left < Right ) 
+    {  /*if there are elements to be sort*/
+        Center = (Left+Right)/2; 
+        MSort(A, TmpArray, Left, Center);   /*T(N/2)*/
+        MSort(A, TmpArray, Center+1, Right);    /*T(N/2)*/
+        Merge(A, TmpArray, Left, Center+1, Right);  /*O(N)*/
+    } 
+} 
+
+void Mergesort( ElementType A[ ], int N ) 
+{   
+    ElementType *TmpArray;  /*need O(N) extra space*/
+    TmpArray = malloc(N*sizeof(ElementType)); 
+    if (TmpArray != NULL) 
+    { 
+        MSort(A, TmpArray, 0, N-1); 
+        free(TmpArray); 
+    } 
+    else FatalError("No space for tmp array!!!"); 
+}
+
    • If a TmpArray is declared locally for each call of Merge, then \(S(N) = O(N\log N)\).
    -
    /*Lpos = start of left half, Rpos = start of right half*/ 
-void Merge( ElementType A[ ], ElementType TmpArray[ ], int Lpos, int Rpos, int RightEnd ) 
-{   
-    int i, LeftEnd, NumElements, TmpPos; 
-    LeftEnd = Rpos-1; 
-    TmpPos = Lpos; 
-    NumElements = RightEnd-Lpos+1; 
-    while( Lpos <= LeftEnd && Rpos <= RightEnd ) /*main loop*/ 
-        if ( A[ Lpos ] <= A[ Rpos ] ) 
-            TmpArray[ TmpPos++ ] = A[ Lpos++ ]; 
-        else 
-            TmpArray[ TmpPos++ ] = A[ Rpos++ ]; 
-    while( Lpos <= LeftEnd ) /*Copy rest of first half*/ 
-        TmpArray[ TmpPos++ ] = A[ Lpos++ ]; 
-    while( Rpos <= RightEnd ) /*Copy rest of second half*/ 
-        TmpArray[ TmpPos++ ] = A[ Rpos++ ]; 
-    for( i = 0; i < NumElements; i++, RightEnd-- ) 
-        /*Copy TmpArray back*/ 
-        A[ RightEnd ] = TmpArray[ RightEnd ]; 
-}
-
    +
    /*Lpos = start of left half, Rpos = start of right half*/ 
+void Merge( ElementType A[ ], ElementType TmpArray[ ], int Lpos, int Rpos, int RightEnd ) 
+{   
+    int i, LeftEnd, NumElements, TmpPos; 
+    LeftEnd = Rpos-1; 
+    TmpPos = Lpos; 
+    NumElements = RightEnd-Lpos+1; 
+    while( Lpos <= LeftEnd && Rpos <= RightEnd ) /*main loop*/ 
+        if ( A[ Lpos ] <= A[ Rpos ] ) 
+            TmpArray[ TmpPos++ ] = A[ Lpos++ ]; 
+        else 
+            TmpArray[ TmpPos++ ] = A[ Rpos++ ]; 
+    while( Lpos <= LeftEnd ) /*Copy rest of first half*/ 
+        TmpArray[ TmpPos++ ] = A[ Lpos++ ]; 
+    while( Rpos <= RightEnd ) /*Copy rest of second half*/ 
+        TmpArray[ TmpPos++ ] = A[ Rpos++ ]; 
+    for( i = 0; i < NumElements; i++, RightEnd-- ) 
+        /*Copy TmpArray back*/ 
+        A[ RightEnd ] = TmpArray[ RightEnd ]; 
+}
+

    Analysis

    \[ T(1)=O(1)\\ @@ -5921,7 +5930,7 @@

    7.7 Quicksort

  • the fastest known sorting algorithm in practice

    • Algorithm

    -
    void Quicksort( ElementType A[ ], int N )
+
    void Quicksort( ElementType A[ ], int N )
 {
     if (N < 2) return;
     pivot = pick any element in A[ ]; 
@@ -5929,7 +5938,7 @@ 
    Algorithm
    
         A1 = { a in S | a <= pivot } and A2 = { a in S | a >= pivot };
     A = Quicksort(A1, N1) and { pivot } and Quicksort(A2, N2);
 }
-
    +
    • The pivot is placed at the right place once and for all.
    • 要研究的问题是如何选取枢纽元和如何划分
      • @@ -5965,56 +5974,56 @@

      Small Arrays

    • Cutoff when \(N\) gets small and use other efficient algorithms (such as insertion sort).

    Implementation

    -
    void Quicksort( ElementType A[ ], int N ) 
-{ 
-    Qsort( A, 0, N-1 ); 
-    /*A:the array*/
-    /*0:Left index*/
-    /*N–1:Right index*/
-}
-
    -
    /* Return median of Left, Center, and Right */ 
-/* Order these and hide the pivot */ 
-ElementType Median3( ElementType A[ ], int Left, int Right ) 
-{
-    int Center = ( Left+Right )/2; 
-    if ( A[ Left ] > A[ Center ] ) 
-        Swap( &A[ Left ], &A[ Center ] ); 
-    if ( A[ Left ] > A[ Right ] ) 
-        Swap( &A[ Left ], &A[ Right ] ); 
-    if ( A[ Center ] > A[ Right ] ) 
-        Swap( &A[ Center ], &A[ Right ] ); 
-    /*Invariant: A[ Left ] <= A[ Center ] <= A[ Right ]*/ 
-    Swap( &A[ Center ], &A[ Right-1 ] ); /*Hide pivot*/ 
-    /*only need to sort A[ Left+1 ] … A[ Right–2 ]*/
-    return A[ Right-1 ];  /*Return pivot*/ 
-}
-
    -
    void Qsort( ElementType A[ ], int Left, int Right ) 
-{
-    int i, j; 
-    ElementType Pivot; 
-    if ( Left + Cutoff <= Right ) 
-    {   /*if the sequence is not too short*/
-        Pivot = Median3( A, Left, Right );  /*select pivot*/
-        i = Left;     
-        j = Right – 1;  /*why not set Left+1 and Right-2?*/
-        for( ; ; ) 
-        { 
-            while ( A[ ++i ] < Pivot ) { }  /*scan from left*/
-            while ( A[ --j ] > Pivot ) { }  /*scan from right*/
-            if ( i < j ) 
-                Swap( &A[ i ], &A[ j ] );  /*adjust partition*/
-            else break;  /*partition done*/
-        } 
-        Swap( &A[ i ], &A[ Right-1 ] ); /*restore pivot */ 
-        Qsort( A, Left, i-1 );    /*recursively sort left part*/
-        Qsort( A, i+1, Right );   /*recursively sort right part*/
-    }  /*end if - the sequence is long*/
-    else /*do an insertion sort on the short subarray*/ 
-        InsertionSort( A+Left, Right-Left+1 );
-}
-
    +
    void Quicksort( ElementType A[ ], int N ) 
+{ 
+    Qsort( A, 0, N-1 ); 
+    /*A:the array*/
+    /*0:Left index*/
+    /*N–1:Right index*/
+}
+
    +
    /* Return median of Left, Center, and Right */ 
+/* Order these and hide the pivot */ 
+ElementType Median3( ElementType A[ ], int Left, int Right ) 
+{
+    int Center = ( Left+Right )/2; 
+    if ( A[ Left ] > A[ Center ] ) 
+        Swap( &A[ Left ], &A[ Center ] ); 
+    if ( A[ Left ] > A[ Right ] ) 
+        Swap( &A[ Left ], &A[ Right ] ); 
+    if ( A[ Center ] > A[ Right ] ) 
+        Swap( &A[ Center ], &A[ Right ] ); 
+    /*Invariant: A[ Left ] <= A[ Center ] <= A[ Right ]*/ 
+    Swap( &A[ Center ], &A[ Right-1 ] ); /*Hide pivot*/ 
+    /*only need to sort A[ Left+1 ] … A[ Right–2 ]*/
+    return A[ Right-1 ];  /*Return pivot*/ 
+}
+
    +
    void Qsort( ElementType A[ ], int Left, int Right ) 
+{
+    int i, j; 
+    ElementType Pivot; 
+    if ( Left + Cutoff <= Right ) 
+    {   /*if the sequence is not too short*/
+        Pivot = Median3( A, Left, Right );  /*select pivot*/
+        i = Left;     
+        j = Right – 1;  /*why not set Left+1 and Right-2?*/
+        for( ; ; ) 
+        { 
+            while ( A[ ++i ] < Pivot ) { }  /*scan from left*/
+            while ( A[ --j ] > Pivot ) { }  /*scan from right*/
+            if ( i < j ) 
+                Swap( &A[ i ], &A[ j ] );  /*adjust partition*/
+            else break;  /*partition done*/
+        } 
+        Swap( &A[ i ], &A[ Right-1 ] ); /*restore pivot */ 
+        Qsort( A, Left, i-1 );    /*recursively sort left part*/
+        Qsort( A, i+1, Right );   /*recursively sort right part*/
+    }  /*end if - the sequence is long*/
+    else /*do an insertion sort on the short subarray*/ 
+        InsertionSort( A+Left, Right-Left+1 );
+}
+

    Note : If set i = Left+1 and j = Right-2, there will be an infinite loop if A[i] = A[j] = pivot.

    @@ -6114,38 +6123,38 @@

    Quickselect

    • 查找第\(K\)最大(最小)元
    -
    /*Places the kth sma11est element in the kth position*/
-/*Because arrays start at 0, this will be index k-1*/
-void Qselect(ElementType A[ ], int k, int Left, int Right)
-{
-    int i, j;
-    ElementType Pivot;
-
-    if (Left + Cutoff <= Right)
-    {
-        Pivot = Median3(A, Left, Right);
-        i = Left; 
-        j = Right-1;
-        for( ; ; )
-        {
-            while(A[ ++i ] < Pivot){ }
-            while(A[ --j ] > Pivot){ }
-            if(i < j)
-                Swap(&A[ i ], &A[ j ]);
-            else
-                break;
-        }
-        Swap(&A[ i ], &A[ Right-1 ]); /*Restore pivot*/
-
-        if(k <= i)
-            Qselect(A, k, Left, i-1);
-        else if (k > i+1)
-            Qselect(A, k, i+1, Right);
-    }
-    else /*Doan insertion sort on the subarray*/
-        InsertionSort(A+Left, Right-Left+1);
-}
-
    +
    /*Places the kth sma11est element in the kth position*/
+/*Because arrays start at 0, this will be index k-1*/
+void Qselect(ElementType A[ ], int k, int Left, int Right)
+{
+    int i, j;
+    ElementType Pivot;
+
+    if (Left + Cutoff <= Right)
+    {
+        Pivot = Median3(A, Left, Right);
+        i = Left; 
+        j = Right-1;
+        for( ; ; )
+        {
+            while(A[ ++i ] < Pivot){ }
+            while(A[ --j ] > Pivot){ }
+            if(i < j)
+                Swap(&A[ i ], &A[ j ]);
+            else
+                break;
+        }
+        Swap(&A[ i ], &A[ Right-1 ]); /*Restore pivot*/
+
+        if(k <= i)
+            Qselect(A, k, Left, i-1);
+        else if (k > i+1)
+            Qselect(A, k, i+1, Right);
+    }
+    else /*Doan insertion sort on the subarray*/
+        InsertionSort(A+Left, Right-Left+1);
+}
+

    image-20210125115951471

    正确答案是D

    @@ -6168,7 +6177,7 @@

    7.10 Bucket Sort

    image-20201221203533117

    -
    Algorithm
+
    Algorithm
 {
     initialize count[ ];
     while(read in a student’s record)
@@ -6178,7 +6187,7 @@ 
    7.10 Bucket Sort
    
         if(count[i]) output list count[i];
     }
 }
-
    +
    \[ T(N,M)=O(M+N) \]
    @@ -6229,80 +6238,80 @@

    8.3 Separate Chaining

    • keep a list of all keys that hash to the same value
    -
    struct ListNode; 
-typedef struct ListNode *Position; 
-struct HashTbl; 
-typedef struct HashTbl *HashTable; 
-struct ListNode { 
-    ElementType Element; 
-    Position Next; 
-}; 
-typedef Position List; 
-/* List *TheList will be an array of lists, allocated later */ 
-/* The lists use headers (for simplicity), */ 
-/* though this wastes space */ 
-struct HashTbl { 
-    int TableSize; 
-    List *TheLists; 
-}; 
-
    +
    struct ListNode; 
+typedef struct ListNode *Position; 
+struct HashTbl; 
+typedef struct HashTbl *HashTable; 
+struct ListNode { 
+    ElementType Element; 
+    Position Next; 
+}; 
+typedef Position List; 
+/* List *TheList will be an array of lists, allocated later */ 
+/* The lists use headers (for simplicity), */ 
+/* though this wastes space */ 
+struct HashTbl { 
+    int TableSize; 
+    List *TheLists; 
+}; 
+

    Create an empty table

    -
    HashTable InitializeTable( int TableSize ) 
-{   
-    HashTable H; 
-    int i; 
-    if ( TableSize < MinTableSize ) 
-    { 
-        Error( "Table size too small" );  
-        return NULL;  
-    } 
-    H = malloc( sizeof( struct HashTbl ) );  /*Allocate table*/
-    if ( H == NULL ) FatalError( "Out of space!!!" ); 
-    H->TableSize = NextPrime( TableSize );  /*Better be prime*/
-    H->TheLists = malloc( sizeof( List )* H->TableSize );  /*Array of lists*/
-    if ( H->TheLists == NULL ) FatalError( "Out of space!!!" );
-    H->TheList = malloc(H->TableSize*sizeof(struct ListNode));
-    for( i = 0; i < H->TableSize; i++ ) 
-    {   /*Allocate list headers*/
-        //H->TheLists[ i ] = malloc( sizeof( struct ListNode ) ); /* Slow! */
-        if ( H->TheLists[ i ] == NULL ) FatalError( "Out of space!!!" ); 
-        else H->TheLists[ i ]->Next = NULL;
-    } 
-    return H; 
-} 
-
    +
    HashTable InitializeTable( int TableSize ) 
+{   
+    HashTable H; 
+    int i; 
+    if ( TableSize < MinTableSize ) 
+    { 
+        Error( "Table size too small" );  
+        return NULL;  
+    } 
+    H = malloc( sizeof( struct HashTbl ) );  /*Allocate table*/
+    if ( H == NULL ) FatalError( "Out of space!!!" ); 
+    H->TableSize = NextPrime( TableSize );  /*Better be prime*/
+    H->TheLists = malloc( sizeof( List )* H->TableSize );  /*Array of lists*/
+    if ( H->TheLists == NULL ) FatalError( "Out of space!!!" );
+    H->TheList = malloc(H->TableSize*sizeof(struct ListNode));
+    for( i = 0; i < H->TableSize; i++ ) 
+    {   /*Allocate list headers*/
+        //H->TheLists[ i ] = malloc( sizeof( struct ListNode ) ); /* Slow! */
+        if ( H->TheLists[ i ] == NULL ) FatalError( "Out of space!!!" ); 
+        else H->TheLists[ i ]->Next = NULL;
+    } 
+    return H; 
+} 
+

    Find a key from a hash table

    -
    Position Find ( ElementType Key, HashTable H ) 
-{ 
-    Position P; 
-    List L; 
-    L = H->TheLists[ Hash( Key, H->TableSize ) ]; 
-    P = L->Next; 
-    while( P != NULL && P->Element != Key )  /*Probably need strcmp*/ 
-        P = P->Next; 
-    return P; 
-} 
-
    +
    Position Find ( ElementType Key, HashTable H ) 
+{ 
+    Position P; 
+    List L; 
+    L = H->TheLists[ Hash( Key, H->TableSize ) ]; 
+    P = L->Next; 
+    while( P != NULL && P->Element != Key )  /*Probably need strcmp*/ 
+        P = P->Next; 
+    return P; 
+} 
+

    Insert a key into a hash table

    -
    void Insert ( ElementType Key, HashTable H ) 
-{ 
-    Position Pos, NewCell; 
-    List L; 
-    Pos = Find( Key, H ); 
-    if ( Pos == NULL ) 
-    {   /*Key is not found, then insert*/
-        NewCell = malloc( sizeof( struct ListNode ) ); 
-        if ( NewCell == NULL ) FatalError( "Out of space!!!" ); 
-        else 
-        { 
-            L = H->TheLists[ Hash( Key, H->TableSize ) ]; /*Compute again is bad*/
-            NewCell->Next = L->Next; 
-            NewCell->Element = Key; /*Probably need strcpy!*/ 
-            L->Next = NewCell; 
-        } 
-    } 
-} 
-
    +
    void Insert ( ElementType Key, HashTable H ) 
+{ 
+    Position Pos, NewCell; 
+    List L; 
+    Pos = Find( Key, H ); 
+    if ( Pos == NULL ) 
+    {   /*Key is not found, then insert*/
+        NewCell = malloc( sizeof( struct ListNode ) ); 
+        if ( NewCell == NULL ) FatalError( "Out of space!!!" ); 
+        else 
+        { 
+            L = H->TheLists[ Hash( Key, H->TableSize ) ]; /*Compute again is bad*/
+            NewCell->Next = L->Next; 
+            NewCell->Element = Key; /*Probably need strcpy!*/ 
+            L->Next = NewCell; 
+        } 
+    } 
+} 
+

    Note : Make the TableSize about as large as the number of keys expected (i.e. to make the loading density factor \(\lambda\approx\)1).

    @@ -6311,7 +6320,7 @@

    8.4 Open Addressing

    • find another empty cell to solve collision(avoiding pointers)
    -
    Algorithm: insert key into an array of hash table
+
    Algorithm: insert key into an array of hash table
 {
     index = hash(key);
     initialize i = 0 ------ the counter of probing;
@@ -6324,7 +6333,7 @@ 
    8.4 Open Addressing
    
     if (table is full) ERROR (“No space left”);
     else insert key at index;
 }
-
    +

    Note : Generally \(\lambda<0.5\).

    @@ -6344,58 +6353,58 @@
    \(4k + 3\), then the quadratic probing \(f(i) = \pm i^2\) can probe the entire table.

    -
    HashTable InitializeTable(int TableSize)
-{
-    HashTable H;
-    int i;
-    if(TableSize < MinTableSize)
-    {
-        Error("Table size too small");
-        return NULL;
-    }
-    /*Allocate table*/
-    H = malloc(sizeof(struct HashTbl));
-    if(H == NULL)
-        Fatal Error("Out of space!!!");
-    H->TableSize = NextPrime(TableSize);
-
-    /*Allocate array of Cells*/
-    H->TheCells = malloc(sizeof(Cell)*H->TableSize);
-    if(H->TheCells == NULL)
-        FatalError("Out of space!!!");
-
-    for(i = 0; i < H->TableSize; i++)
-        H->TheCells[ i ].Info = Empty;
-    return H;
-}
-
    -
    Position Find(ElementType Key, HashTable H) 
-{   
-    Position CurrentPos; 
-    int CollisionNum; 
-    CollisionNum = 0; 
-    CurrentPos = Hash(Key, H->TableSize); 
-    while(H->TheCells[ CurrentPos ].Info != Empty &&
-          H->TheCells[ CurrentPos ].Element != Key) 
-    { 
-        CurrentPos += 2*++CollisionNum-1; 
-        if (CurrentPos >= H->TableSize)  
-            CurrentPos -= H->TableSize;   /*Faster than mod*/
-    } 
-    return CurrentPos; 
-} 
-
    -
    void Insert(ElementType Key, HashTable H) 
-{ 
-    Position Pos; 
-    Pos = Find(Key, H); 
-    if (H->TheCells[ Pos ].Info != Legitimate) 
-    { /*OK to insert here*/ 
-        H->TheCells[ Pos ].Info = Legitimate; 
-        H->TheCells[ Pos ].Element = Key; /*Probably need strcpy*/ 
-    } 
-} 
-
    +
    HashTable InitializeTable(int TableSize)
+{
+    HashTable H;
+    int i;
+    if(TableSize < MinTableSize)
+    {
+        Error("Table size too small");
+        return NULL;
+    }
+    /*Allocate table*/
+    H = malloc(sizeof(struct HashTbl));
+    if(H == NULL)
+        Fatal Error("Out of space!!!");
+    H->TableSize = NextPrime(TableSize);
+
+    /*Allocate array of Cells*/
+    H->TheCells = malloc(sizeof(Cell)*H->TableSize);
+    if(H->TheCells == NULL)
+        FatalError("Out of space!!!");
+
+    for(i = 0; i < H->TableSize; i++)
+        H->TheCells[ i ].Info = Empty;
+    return H;
+}
+
    +
    Position Find(ElementType Key, HashTable H) 
+{   
+    Position CurrentPos; 
+    int CollisionNum; 
+    CollisionNum = 0; 
+    CurrentPos = Hash(Key, H->TableSize); 
+    while(H->TheCells[ CurrentPos ].Info != Empty &&
+          H->TheCells[ CurrentPos ].Element != Key) 
+    { 
+        CurrentPos += 2*++CollisionNum-1; 
+        if (CurrentPos >= H->TableSize)  
+            CurrentPos -= H->TableSize;   /*Faster than mod*/
+    } 
+    return CurrentPos; 
+} 
+
    +
    void Insert(ElementType Key, HashTable H) 
+{ 
+    Position Pos; 
+    Pos = Find(Key, H); 
+    if (H->TheCells[ Pos ].Info != Legitimate) 
+    { /*OK to insert here*/ 
+        H->TheCells[ Pos ].Info = Legitimate; 
+        H->TheCells[ Pos ].Element = Key; /*Probably need strcpy*/ 
+    } 
+} 
+

    Note :

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