Graphs/Floyd Warshall
From charlesreid1
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 $ 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) )
References
Related
Graphs/Transitive Closure - the transitive closure of a directed graph can be found using the Floyd Warshall algorithm
Graphs/Dijstra - related algorithm for finding the shortest paths between two vertices in a graph
Links
Link to Floyd Warshall entry in the NIST algorithms and data structures dictionary: https://xlinux.nist.gov/dads/HTML/floydWarshall.html
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