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johnson_all_pairs_shortest_paths

```// named parameter version
template <class VertexAndEdgeListGraph, class DistanceMatrix,
class VertexID, class Weight, class BinaryPredicate,
class BinaryFunction, class Infinity, class DistanceZero>
bool
johnson_all_pairs_shortest_paths(VertexAndEdgeListGraph& g1,
DistanceMatrix& D,
VertexID id1, Weight w1, const BinaryPredicate& compare,
const BinaryFunction& combine, const Infinity& inf,
DistanceZero zero);

template <typename Graph, typename DistanceMatrix, typename P, typename T, typename R>
bool johnson_all_pairs_shortest_paths(Graph& g, DistanceMatrix& D,
const bgl_named_params<P, T, R>& params = all defaults)

// non-named parameter version
template <typename Graph, typename DistanceMatrix,
typename VertexIndex, typename WeightMap, typename DT>
bool
johnson_all_pairs_shortest_paths(VertexAndEdgeListGraph& g1,
DistanceMatrix& D,
VertexIndex i_map, WeightMap w_map, DT zero)
```

This algorithm finds the shortest distance between every pair of vertices in the graph. The algorithm returns false if there is a negative weight cycle in the graph and true otherwise. The distance between each pair of vertices is stored in the distance matrix D. This is one of the more time intensive graph algorithms, having a time complexity of O(V E log V).

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

Parameters

IN: Graph& g
A directed or undirected graph. The graph type must be a model of Vertex List Graph, Edge List Graph, and Incidence Graph.
OUT: DistanceMatrix& D
The length of the shortest path between each pair of vertices u,v in the graph is stored in D[u][v]. The tuple of types (DistanceMatrix, vertices_size_type, D) must be a model of BasicMatrix where D is the value type of the DistanceMap. There must be implicit conversions between the value type of the distance matrix and the value type of the weight map.

Named Parameters

IN: weight_map(WeightMap w_map)
The weight or "length" of each edge in the graph. The type 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 support a subtraction operation.
Default: get(edge_weight, g)
UTIL: weight_map2(WeightMap2 w2_map)
This parameter is no longer needed, and will be ignored.
IN: vertex_index_map(VertexIndexMap i_map)
This maps each vertex to an integer in the range [0, num_vertices(g)). This is necessary for efficient updates of the heap data structure in the internal call to Dijkstra's algorithm. The type VertexIndexMap must be a model of Readable Property Map. The value type of the map must be an integer type. The vertex descriptor type of the graph needs to be usable as the key type of the map.
Default: get(vertex_index, g) Note: if you use this default, make sure your graph has an internal vertex_index property. For example, adjacency_list with VertexList=listS does not have an internal vertex_index property.
UTIL: distance_map(DistanceMap d_map)
This parameter is no longer needed, and will be ignored.
IN: distance_compare(CompareFunction cmp)
This function is use to compare distances to determine which vertex is closer to the source vertex. The CompareFunction type must be a model of Binary Predicate and have argument types that match the value type of the WeightMap property map.
Default: std::less<DT> with DT=typename property_traits<WeightMap>::value_type
IN: distance_combine(CombineFunction cmb)
This function is used to combine distances to compute the distance of a path. The CombineFunction type must be a model of Binary Function. Both argument types and the return type of the binary function must match the value type of the WeightMap property map. This operation is required to act as the sum operation for the weight type; in particular, it must be the inverse of the binary - operator on that type.
Default: std::plus<DT> with DT=typename property_traits<WeightMap>::value_type
IN: distance_inf(DT inf)
This value is used to initialize the distance for each vertex before the start of the algorithm. The type DT must be the value type of the WeightMap.
Default: std::numeric_limits::max()
IN: distance_zero(DT zero)
This value is used to initialize the distance for the source vertex before the start of the algorithm. The type DT must be the value type of the WeightMap.
Default: 0
UTIL/OUT: color_map(ColorMap c_map)
This is used during the execution of the algorithm to mark the vertices. The vertices start out white and become gray when they are inserted in the queue. They then turn black when they are removed from the queue. At the end of the algorithm, vertices reachable from the source vertex will have been colored black. All other vertices will still be white. The type ColorMap must be a model of Read/Write Property Map. A vertex descriptor must be usable as the key type of the map, and the value type of the map must be a model of Color Value.
Default: an iterator_property_map created from a std::vector of default_color_type of size num_vertices(g) and using the i_map for the index map.

Complexity

The time complexity is O(V E log V).

Example

The file example/johnson-eg.cpp applies Johnson's algorithm for all-pairs shortest paths to the example graph from page 568 of the CLR [8].