# Notes

The Floyd-Warshall algorithm is an algorithm for computing the transitive closure G* of a graph G.

More generally, we can think of it as an algorithm for finding shortest paths in a graph. It works for weighted graphs, which can have positive or negative weights (unlike the Graphs/Dijkstra algorithm, which does not allow negative weights), but it cannot have negative weight cycles (that is, a cycle whose weights sum to a negative value).

The Floyd Warshall algorithm will find the lengths of the shortest paths between all pairs of vertices in ${\displaystyle O(n^{3})\sim O(|V|^{3})}$ time.

The core idea of the algorithm is finding a shortest path from vertex i to vertex j that only utilizes a set of vertices labeled 1 to k. This is denoted shortestPath(i,j,k).

The base case, of k being 0, is when i and j are directly connected. Then the shortest path from i to j that does not pass through any other vertices is denoted:

shortestPath(i,j,0) = w(i,j)

where w(i,j) denotes the weight of the edge connecting vertex i to vertex j. From there, we can construct a new shortest path by taking the minimum of the shortest path passing through k, and the shortest path that goes from i to k-1 and then from k-1 to j:

shortestPath(i,j,k-1) = min( shortestPath(i,j,k) , shortestPath(i,k-1,j) + shortestPath(k-1,j,k) )

## Pseudocode

```def floyd_warshall(g):
vertices = get list of vertices from g
n = length of vertices
gstar[0] = deep copy of g
for k = 1..n:
gstar[k] = deep copy of g[k-1]
for i in 1..n:
skip i=j and i=k
for j in 1..n:
skip j=k
if edge(vertices[i],vertices[k]) in gstar[k-1] and edge(vertices[k],vertices[j]) in gstar[k-1]:
add edge(vertices[i],vertices[j]) to gstar[k]
return gstar[n]
```

## Big O Cost

The cost of the Floyd Warshall algorithm is ${\displaystyle O(n^{3})}$

This is asymptotically the same as running a DFS on every node. However, this algorithm performs better for dense graphs or graphs represented with an adjacency matrix data structure.

# References

## Related

Graphs/Transitive Closure - the transitive closure of a directed graph can be found using the Floyd Warshall algorithm

Graphs/Dijkstra - related algorithm for finding the shortest paths between two vertices in a graph