The Floyd-Warshall algorithm transmits the closure and the floyd-warshall algorithm at any two points.

The Floyd-Warshall algorithm transmits the closure and the floyd-warshall algorithm at any two points.

The Floyd-Warshall algorithm is a powerful tool for solving the shortest of any two points. It is also applicable to the presence of negative edges.

DP concept. Assume that the shortest distance from I to j is d [K] [I] [j].
Then, the first K + 1 points can be divided into two situations.
① K + 1 point is used for the shortest circuit from I to j (d [k + 1] [I] [j] = d [k] [I] [j])
② The shortest path from I to j does not use the K + 1 point (d [k + 1] [I] [j] = d [k] [I] [k] + d [k] [k] [j]);
So,D [k + 1] [I] [j] = min (d [k] [I] [j], d [k] [I] [k] + d [k] [k] [j])

You can use a scrolling array to write a two-dimensional array.
D [I] [j] = min (d [I] [j], d [I] [k] + d [k] [j])

The Floyd algorithm can be solved in O (V ^ 3) Time to determine whether a negative circle exists in the graph, you only need to check whether d [I] [I] is a negative vertex I.

Code

// D [I] [j] indicates the weight of I-> j. If no value exists, it is set to INF, but d [I] [I] is set to 0for (int k = 0; k <V; k ++) {for (int I = 0; I <V; I ++) {for (int j = 0; j <V; j ++) {d [I] [j] = min (d [I] [j], d [I] [k] + d [k] [j]) ;}}

Because the implementation is very simple, if the complexity is within the tolerable range, the Floyd-Warshall algorithm can also be used for solving the single-source shortest.

In a directed graph, you can use the Floyd algorithm to determine whether a path exists for each two points.Passing Closure.
You only need to change itD [I] [j] = d [I] [j] | (d [I] [k] & d [k] [j])(Preprocessing also needs to change)