Floyd-Warshall All Pairs Shortest Paths

# `floyd_warshall_all_pairs_shortest_paths`

    // Named parameters version
    template <class VertexListGraph, class DistanceMatrix,
        class P, class T, class R>
    bool floyd_warshall_initialized_all_pairs_shortest_paths(
        const VertexListGraph& g, DistanceMatrix& d,
        const bgl_named_params<P, T, R>& params)

    template <class VertexAndEdgeListGraph, class DistanceMatrix,
        class P, class T, class R>
    bool floyd_warshall_all_pairs_shortest_paths(
        const VertexAndEdgeListGraph& g, DistanceMatrix& d,
        const bgl_named_params<P, T, R>& params)

    // Positional parameter versions
    \begin{verbatim}
    template <typename VertexListGraph, typename DistanceMatrix,
        typename BinaryPredicate, typename BinaryFunction,
        typename Infinity, typename Zero>
    bool floyd_warshall_initialized_all_pairs_shortest_paths(
        const VertexListGraph& g, DistanceMatrix& d,
        const BinaryPredicate& compare, const BinaryFunction& combine,
        const Infinity& inf, const Zero& zero)

    template <typename VertexAndEdgeListGraph, typename DistanceMatrix,
        typename WeightMap, typename BinaryPredicate,
        typename BinaryFunction, typename Infinity, typename Zero>
    bool floyd_warshall_all_pairs_shortest_paths(
        const VertexAndEdgeListGraph& g, DistanceMatrix& d,
        const WeightMap& w, const BinaryPredicate& compare,
        const BinaryFunction& combine,
        const Infinity& inf, const Zero& zero)

These algorithms find the shortest distance between every pair of vertices in the graph. The algorithms return false if there is a negative weight cycle in the graph, true otherwise. The shortest distance between each pair of vertices is stored in the distance matrix `d`. The difference between the two algorithms is in whether the distance matrix is assumed to be initialized or not, as discussed below under the OUT parameter description.

This algorithm should be used to compute shortest paths between every pair of vertices for dense graphs. For sparse graphs, use `johnson_all_pairs_shortest_paths`.

### Where Defined

`boost/graph/floyd_warshall_shortest.hpp`

### Parameters

IN: `Graph& g`

> A directed or undirected graph. The graph must be a model of Vertex List Graph for calls to `floyd_warshall_initialized_all_pairs_shortest_paths`, and Vertex And Edge List Graph for calls to `floyd_warshall_all_pairs_shortest_paths`.

OUT: `DistanceMatrix& d`

> The length of the shortest path between each pair of vertices `u`,`v` are stored in the matrix at location `D[u][v]`. The DistanceMatrix must be of type `{M, I, V}` where `I` is of type `vertex_descriptor` and `V` is the type of the `weight_map`. The set of types must be a model of BasicMatrix, with the exceptions that it isn't required to run in constant time, and it must be mutable. The matrix must be properly initialized when it is passed to the function `floyd_warshall_initialized_all_pairs_shortest_paths`. If the function `floyd_warshall_all_pairs_shortest_paths` is used then the matrix will be initialized for the user.

### Named Parameters

IN: `weight_map(WeightMap w)`

> The weight of length of each edge in the graph. The `WeightMap` must be a model of Readable Property Map. The edge descriptor type of the graph needs to be usable as the key type for the weight map. The `value_type` of the weight map must be the type of the `DistanceMatrix`, and must always either be part of the graph passed to the function, or passed in as a parameter.
> Default: `get(edge_weight, g)`

IN: `distance_compare(CompareFunction cmp)`

> The function used to compare distances to determine which target vertex is closer to the source vertex. The CompareFunction must be a model of `BinaryPredicate`. The argument types must match the value type of the `WeightMap`.
> Default: `std::less<WM>`with `WM = typename property_traits<WeightMap>::value_type`

IN: `distance_combine(CombineFunction cmb)`

> The function used to combine distance to compute the distance of a path. The CombineFunction must be a model of `BinaryFunction`. The argument types must match the value type of the `WeightMap`. The result type must be the same as the distance value type.
> Default: `std::plus<WM>` with `WM = typename property_traits<WeightMap>::value_type`

IN: `distance_inf(WM inf)`

> The value used to initialize the distance for each vertex before starting the algorithm, and to represent the distance between vertices for which there is not path. Should be larger than any possible valid path length. The argument type must match the value type of the `WeightMap`.
> Default: `std::numeric_limits<WM>::max()` with `WM = typename property_traits<WeightMap>::value_type`

IN: `distance_zero(WM zero)`

> The value used to represent the distance from a vertex to itself, and to determine if a value is negative. The argument type must match the value type of the `WeightMap`.
> Default: `0`

### Complexity

The time complexity is O(V3).

Copyright © 2002-2004 | Lauren Foutz, Rensselaer Polytechnic Institute
| Scott Hill, Rensselaer Polytechnic Institute