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题目地址

https://leetcode.com/problems/combination-sum/description/

题目描述

Given a set of candidate numbers (candidates) (without duplicates) and a target number (target), find all unique combinations in candidates where the candidate numbers sums to target.

The same repeated number may be chosen from candidates unlimited number of times.

Note:

All numbers (including target) will be positive integers.
The solution set must not contain duplicate combinations.
Example 1:

Input: candidates = [2,3,6,7], target = 7,
A solution set is:
[
  [7],
  [2,2,3]
]
Example 2:

Input: candidates = [2,3,5], target = 8,
A solution set is:
[
  [2,2,2,2],
  [2,3,3],
  [3,5]
]

前置知识

  • 回溯法

思路

这道题目是求集合，并不是求极值，因此动态规划不是特别切合，因此我们需要考虑别的方法。

这种题目其实有一个通用的解法，就是回溯法。 网上也有大神给出了这种回溯法解题的 通用写法，这里的所有的解法使用通用方法解答。 除了这道题目还有很多其他题目可以用这种通用解法，具体的题目见后方相关题目部分。

我们先来看下通用解法的解题思路，我画了一张图：

通用写法的具体代码见下方代码区。

关键点解析

  • 回溯法
  • backtrack 解题公式

代码

  • 语言支持: Javascript，Python3
/*
 * @lc app=leetcode id=39 lang=javascript
 *
 * [39] Combination Sum
 *
 * https://leetcode.com/problems/combination-sum/description/
 *
 * algorithms
 * Medium (46.89%)
 * Total Accepted:    326.7K
 * Total Submissions: 684.2K
 * Testcase Example:  '[2,3,6,7]\n7'
 *
 * Given a set of candidate numbers (candidates) (without duplicates) and a
 * target number (target), find all unique combinations in candidates where the
 * candidate numbers sums to target.
 *
 * The same repeated number may be chosen from candidates unlimited number of
 * times.
 *
 * Note:
 *
 *
 * All numbers (including target) will be positive integers.
 * The solution set must not contain duplicate combinations.
 *
 *
 * Example 1:
 *
 *
 * Input: candidates = [2,3,6,7], target = 7,
 * A solution set is:
 * [
 * ⁠ [7],
 * ⁠ [2,2,3]
 * ]
 *
 *
 * Example 2:
 *
 *
 * Input: candidates = [2,3,5], target = 8,
 * A solution set is:
 * [
 * [2,2,2,2],
 * [2,3,3],
 * [3,5]
 * ]
 *
 */

function backtrack(list, tempList, nums, remain, start) {
  if (remain < 0) return;
  else if (remain === 0) return list.push([...tempList]);
  for (let i = start; i < nums.length; i++) {
    tempList.push(nums[i]);
    backtrack(list, tempList, nums, remain - nums[i], i); // 数字可以重复使用， i + 1代表不可以重复利用
    tempList.pop();
  }
}
/**
 * @param {number[]} candidates
 * @param {number} target
 * @return {number[][]}
 */
var combinationSum = function(candidates, target) {
  const list = [];
  backtrack(list, [], candidates.sort((a, b) => a - b), target, 0);
  return list;
};

Python3 Code:

class Solution:
    def combinationSum(self, candidates: List[int], target: int) -> List[List[int]]:
        """
        回溯法，层层递减，得到符合条件的路径就加入结果集中，超出则剪枝；
        主要是要注意一些细节，避免重复等；
        """
        size = len(candidates)
        if size <= 0:
            return []
        
        # 先排序，便于后面剪枝
        candidates.sort()
        
        path = []
        res = []
        self._find_path(target, path, res, candidates, 0, size)
        
        return res
        
    def _find_path(self, target, path, res, candidates, begin, size):
        """沿着路径往下走"""
        if target == 0:
            res.append(path.copy())
        else:
            for i in range(begin, size):
                left_num = target - candidates[i]
                # 如果剩余值为负数，说明超过了，剪枝
                if left_num < 0:
                    break
                # 否则把当前值加入路径
                path.append(candidates[i])
                # 为避免重复解，我们把比当前值小的参数也从下一次寻找中剔除，
                # 因为根据他们得出的解一定在之前就找到过了
                self._find_path(left_num, path, res, candidates, i, size)
                # 记得把当前值移出路径，才能进入下一个值的路径
                path.pop()

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