Update Floyd-Warshall READMEs.

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)