Update Floyd-Warshall READMEs.

Author: trekhleb

README.md

@@ -154,12 +154,13 @@ algorithm is an abstraction higher than a computer program.
   * `A` [Levenshtein Distance](src/algorithms/string/levenshtein-distance) - minimum edit distance between two sequences
   * `A` [Longest Common Subsequence](src/algorithms/sets/longest-common-subsequence) (LCS)
   * `A` [Longest Common Substring](src/algorithms/string/longest-common-substring)
-  * `A` [Longest Increasing subsequence](src/algorithms/sets/longest-increasing-subsequence)
+  * `A` [Longest Increasing Subsequence](src/algorithms/sets/longest-increasing-subsequence)
   * `A` [Shortest Common Supersequence](src/algorithms/sets/shortest-common-supersequence)
   * `A` [0/1 Knapsack Problem](src/algorithms/sets/knapsack-problem)
   * `A` [Integer Partition](src/algorithms/math/integer-partition)
   * `A` [Maximum Subarray](src/algorithms/sets/maximum-subarray)
   * `A` [Bellman-Ford Algorithm](src/algorithms/graph/bellman-ford) - finding shortest path to all graph vertices
+  * `A` [Floyd-Warshall Algorithm](src/algorithms/graph/floyd-warshall) - find shortest paths between all pairs of vertices
   * `A` [Regular Expression Matching](src/algorithms/string/regular-expression-matching)
 * **Backtracking** - similarly to brute force, try to generate all possible solutions, but each time you generate next solution you test
 if it satisfies all conditions, and only then continue generating subsequent solutions. Otherwise, backtrack, and go on a

src/algorithms/graph/floyd-warshall/README.md

@@ -1,5 +1,94 @@
-# Floyd–Warshall algorithm
+# Floyd–Warshall Algorithm
+
+In computer science, the **Floyd–Warshall algorithm** is an algorithm for finding
+shortest paths in a weighted graph with positive or negative edge weights (but
+with no negative cycles). A single execution of the algorithm will find the 
+lengths (summed weights) of shortest paths between all pairs of vertices. Although 
+it does not return details of the paths themselves, it is possible to reconstruct 
+the paths with simple modifications to the algorithm.
+
+## Algorithm
+
+The Floyd–Warshall algorithm compares all possible paths through the graph between 
+each pair of vertices. It is able to do this with `O(|V|^3)` comparisons in a graph.
+This is remarkable considering that there may be up to `|V|^2` edges in the graph, 
+and every combination of edges is tested. It does so by incrementally improving an 
+estimate on the shortest path between two vertices, until the estimate is optimal.
+
+Consider a graph `G` with vertices `V` numbered `1` through `N`. Further consider 
+a function `shortestPath(i, j, k)` that returns the shortest possible path 
+from `i` to `j` using vertices only from the set `{1, 2, ..., k}` as 
+intermediate points along the way. Now, given this function, our goal is to 
+find the shortest path from each `i` to each `j` using only vertices 
+in `{1, 2, ..., N}`.
+
+![Recursive Formula](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9b75e25063384ccca499c56f9a279abf661ad3b)
+
+![Recursive Formula](https://wikimedia.org/api/rest_v1/media/math/render/svg/34ac7c89bbb18df3fd660225fd38997079e5e513)
+![Recursive Formula](https://wikimedia.org/api/rest_v1/media/math/render/svg/0326d6c14def89269c029da59eba012d0f2edc9d)
+
+This formula is the heart of the Floyd–Warshall algorithm.
+
+## Example
+
+The algorithm above is executed on the graph on the left below:
+
+![Example](https://upload.wikimedia.org/wikipedia/commons/2/2e/Floyd-Warshall_example.svg)
+
+In the tables below `i` is row numbers and `j` is column numbers.
+
+
+**k = 0**
+
+|       | 1   | 2   | 3   | 4   |
+|:-----:|:---:|:---:|:---:|:---:|
+| **1** |	0   |	∞   |	−2  | ∞   |
+| **2** |	4   |	0   |	3	  | ∞   |
+| **3** |	∞   |	∞   |	0	  | 2   |
+| **4** |	∞   |	−1  | ∞   | 0   |
+
+
+**k = 1**
+
+|       | 1   | 2   | 3   | 4   |
+|:-----:|:---:|:---:|:---:|:---:|
+| **1** | 0   | ∞   | −2  | ∞   |
+| **2** | 4   | 0   |  2  | ∞   |
+| **3** | ∞   | ∞   |  0  | 2   |
+| **4** | ∞   | −   |  ∞  | 0   |
+
+
+**k = 2**
+
+|       | 1   | 2   | 3   | 4   |
+|:-----:|:---:|:---:|:---:|:---:|
+| **1** |	0   |	∞   |	−2  | ∞   |
+| **2** |	4   |	0   | 2	  | ∞   |
+| **3** |	∞   |	∞	  | 0	  | 2   |
+| **4** |	3   |	−1  | 1   | 0   |
+
+
+**k = 3**
+
+|       | 1   | 2   | 3   | 4   |
+|:-----:|:---:|:---:|:---:|:---:|
+| **1** |	0   |	∞   |	−2  | 0   |
+| **2** |	4   |	0   |	2	  | 4   |
+| **3** |	∞   |	∞   |	0	  | 2   |
+| **4** |	3   |	−1  | 1   | 0   |
+
+
+**k = 4**
+
+|       | 1   | 2   | 3   | 4   |
+|:-----:|:---:|:---:|:---:|:---:|
+| **1** |	0   |	−1  | −2  | 0   |
+| **2** |	4   |	0	  | 2	  | 4   |
+| **3** |	5   |	1	  | 0	  | 2   |
+| **4** |	3   |	−1  | 1   | 0   |
 
 ## References
 
 - [Wikipedia](https://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithm)
+- [YouTube (by Abdul Bari)](https://www.youtube.com/watch?v=oNI0rf2P9gE&list=PLLXdhg_r2hKA7DPDsunoDZ-Z769jWn4R8&index=74)
+- [YouTube (by Tushar Roy)](https://www.youtube.com/watch?v=LwJdNfdLF9s&list=PLLXdhg_r2hKA7DPDsunoDZ-Z769jWn4R8&index=75)