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