docs(roadsAndLibraries): update and proofread README.md

src/algorithm_practice/Datastructure_Algorithms/Graph/roadsAndLibraries/README.md

@@ -1,84 +1,92 @@
-Source: https://www.hackerrank.com/challenges/torque-and-development/
+# Roads and Libraries
 
-TODO(domfarolino): revise this post.
+Source: https://www.hackerrank.com/challenges/torque-and-development/
 
 This is a pretty interesting graph problem. It vexed me for a bit until I made some cruicial realizations.
 
 # Divide the problem into connected components
 
-When starting with this problem I fumbled around quite a bit. Eventually I came to some good realizations:
+When starting with this problem I fumbled around quite a bit, but eventually I came to some realizations revolving
+around focusing on each connected component in the given graph:
 
- - We'll need at least one library per connected component
- - In each component, there are two extremes:
+ - We'll need at least one library per connected component in the graph
+ - In each connected component, there are two extremes:
    - Every city in a connected component has a library
    - Only one city in a connected component has a library
 
-My next thought was that the naive solution would be to find all possible combinations of library/road
-allocations in between the extremes, which seems combinatorially explosive. For example, what if there
-were not the extreme `n` libraries and `0` roads in a component, but instead `n - 1` libraries and `1`
-road, or `n - 2` libraries and `2` roads. How many different ways can we
-[*choose*](https://en.wikipedia.org/wiki/Binomial_coefficient) how to allocate which cities have libraries
-and which cities to connect, and more importantly, does the choosing of these actually affect the cost?
-Determining the number of possible choices we can make when allocating libraries to cities is actually pretty
-easy (it's just the summation of binomial coefficients, [see here](https://math.stackexchange.com/questions/519832/)),
-it would just be combinatorially explosive to go through each one; was it necessary?
+My next thought was that the naïve solution would be to find all possible combinations of allocated libraries
+and built roads between the aforementioned extremes, which seems combinatorially explosive. For example, what
+if there were not the extreme `n` libraries and `0` roads in a component, but instead `n - 1` libraries and `1`
+road, or maybe `n - 2` libraries and `2` roads. It is obvious to determine how many different ways we can
+[*choose*](https://en.wikipedia.org/wiki/Binomial_coefficient) to allocate `n - k` libraries amongst `n` cities
+in a component. But how many different ways can we decide what roads to build given a particular allocation? Do
+the different choices of which roads to build for some given library allocation actually affect the cost?
+
+When looking at an example graph with five (once-) connected cities, I realized that the allocation of libraries
+doesn't matter at all and won't affect the cost. (I had considered the idea that perhaps the degree of each city
+might have an affect on, or indicate priority of library assignment). Which cities we choose to build libraries
+in is irrelevant as long as we don't waste a road connecting two library-bearing cities when we could use it to
+connect a non-library-bearing city to a library-bearing one.
 
-When looking at an example graph with five (once-) connected cities I realized that the allocation of libraries
-doesn't matter at all and won't affect the cost. (I was considering the idea that perhaps the degree of each city
-might have an affect on, or indicate priority of library assignment). The allocation makes no difference as long
-as we don't waste a road connecting two library-bearing cities, because why would we do that?
+> Warning, entering a tangent:
 
-[Enter a tangent]...
+![entering a tagent](https://apps.azdot.gov/files/traffic/moas/thumbgif/w/w01-010.gif)
 
 The whole reason this accidentally-connecting-two-library-bearing-cities issue came up is because I was examining a
-quite feasible 5-city graph with a cycle trying to allocate `3` libraries and `2` roads. I wondered if I could choose
-a "bad" allocation of libraries and roads, namely one that doesn't actually connect each city in the component. This is
-certainly possible in a graph with cycles when only dealing with `numberOfCities` resources (`3` libraries and `2` roads).
+quite feasible 5-city graph with a cycle, trying to allocate `3` libraries and `2` roads. I wondered what would happen
+if I chose to waste the building of a road on connecting two cities that both bore libraries, instead of building a road
+from a city that was not connected to a library to a city in the same component that was. This "bad allocation" can happen
+because of this cycle (effectively wasting a road on a group of cities that don't need another built).
 
-I was then worried about making sure my implementation would not accidentally theoretically waste a road on two
-library-bearing cities, and then I realized well yeah, if the allocation doesn't matter, we just have to know that
-some working allocation exists, and that will be the minimum total cost for such choices of the number of libraries
-and roads for that connected component.
+I was wondering how I could ensure my implementation would not accidentally do this, but then I realized it wouldn't be
+necessary. *Some* proper allocation of built roads exists, and as long as I know it exists, I don't have to worry about
+accidentally choosing a "bad allocation", because I'll be using the same number of roads either way! And I know that by
+using that number of roads, it *is* possible to connect all cities, therefore the exact layout is meaningless.
 
 # A connected component is at least a tree
 
-The "choice" of which roads to build dissolves when you realize that the connected component by definition is at least a
-tree, and thus always has valid allocations of libraries and roads in the form of:
+The reason there is definitely *a* working allocation of roads to build is true is because we can ignore cycles that would
+otherwise form a "bad allocation", because the connected component contains *at least* the roads necessary to connect all
+cities without roads that form cycles. In other words, the connected component is at least a tree, so mathematically *some*
+non-wasteful layout of roads exists.
+
+A group of connectable cities can therefore have all of its cities connected to libraries in `N` different ways, where `N`
+is the number of cities in the group:
 
 `N - K` libraries + `K` roads, `∀ K < N` (remember, we need at least one library).
 
-This means each connected component had `N` possible solutions, and for each of the values of `K`, we needed to choose the
-minumum one. Going through some examples I realized the best answer always seemed to be one of the extreme allocations, namely
-an allocation with all `N` libraries or only `1` library. I tried to find an example where one of the middleground less
-extreme allocations could be more optimal, but I came to the conclusion that that will never be the case, because we greedily
-want to choose to employ as many of the cheapest resource (either libraries or roads) as possible. In other words, if roads were
-cheaper to build then libraries, and there exist the possible roads to repair to connect the entire component (the definition!),
-then we'd want to only build `1` library, and as many remaining roads as we'd need. We could build two libraries, and one less
-road, but that would give us the same connected result but with a higher cost, unnecessarily.
+This means each connected component has `N` possible solutions, and for each of the values of `K` (ranging from `0` to `N - 1`),
+we need to choose the most cost-efficient one. Going through some examples I realized the best answer always seemed to be one of
+the extremes, namely a layout containing all `N` libraries and `0` roads, or only `1` library and `N - 1` roads. Trying to find
+an example in which one of the middleground distributions was more optimal, I eventually came to the conclusino that this will never
+be the case, because we greedily want to choose to employ as many of the cheapest resource (libraries or roads) as possible!
+
+In other words, if roads were cheaper to build then libraries, and there exist the possible roads to repair to connect the entire
+component (the definition!), then we'd want to only build `1` library and as many remaining roads as we'd need. We could build two
+libraries, and one less road, but that would give us the same connected result but with a higher cost, unnecessarily.
 
 # Implementation design
 
 When thinking about the implementation, I knew the number of connected components was relevant to this problem. I also knew
-we could get an entire connected component (but more importantly its size) using a trivial-to-implement BFS algorithm. I figured
+we could get an entire connected component (but more importantly, its size) using good ole BFS (DFS suffices too). I figured
 I'd use an adjecency list to store the graph, since I wasn't going to perform any operations that a matrix would be more suited
 for. The necessary steps were something like this:
 
  - Build the graph's adjacency list
- - For each connected component
+ - For each connected component:
    - Get the size of the component
-   - Minimal cost of connecting this component was `min(a, b)` where:
-     - `a = numCities * costLib`
-     - `b = costLib + (numCities - 1) * costRoad`
-   - With the minimal cost of the component in hand, add the value to the running some, and perform the same operation for the next component.
+   - Compute the minimal cost of connecting this component (rebuilding existing roads), which is `min(a, b)` where:
+     - `a = componentSize * costLib`
+     - `b = costLib + (componentSize - 1) * costRoad`
+   - With the minimal cost of the component in hand, add the value to a running sum, and perform the same operation for the next component.
 
-Moving from component-to-component is as easy as just using BFS with some sort of global visitation store.
+Moving from component-to-component is as easy as just using BFS with some sort of global visitation data structure.
 We can try to find a connected component from each given city. The first time we run BFS, we'll mark *all* nodes in
 the discovered component as visited. Then in the next given city, we'll try to find another connected component *if*
-the city has not already been visited (does not exist as a part of an already-discovered connected component). We keep
-a running sum, adding to it the minimum cost required to connect a once-connected component, and eventually return the
-final value.
+the city has not already been visited (does not exist as a part of an already-discovered connected component). Eventually
+we'll return the value of a running sum we've kept (as mentioned above).
 
-Time complexity: O(n) (by marking nodes as visited, we're repeating ourselves)
+Time complexity: O(n)
 Space complexity: O(n)
 
 *It should be noted that the complexity of this algorithm could easily by O(n^2) (due to edge processing in the complete