Floyd Warshall Algorithm


This algorithm runs in O(n^3) time yet it gives shortest path between any two points. The algorithms simply looks the shortest path between two nodes passing a selected node. This operation takes place by using an adjacency matrix. Where the graph is represented using a matrix. Down the rows and through the columns nodes are represented. In the matrix distances are displayed. Distances are taken from row number to columns. See the following example graph with 3 nodes

this contains edges 0-2 = 5 and 1-2 = 10 only. Such a graph will be shown as 

Graph = {
{  0    INF   5},
{INF   0    10},
{INF INF   0}

Here we consider Rows as the source vertex and Columns as the target vertex.


The algorithm consider a vertex and calculate all the paths through it.
In a vector notation this makes more sense. Say we have a path P(a,b) path a to b. What we do in the algorithm is consider all paths P(a,c) + P(c,b) = P(a,b) until we get min(P(a,b)). This can be implemented using 3 for(;;) loops as follows. With an adjacency matrix the work is so simple. Once done taking the path between any two points becomes peanuts.

Java Code

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