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Johnson All Pairs Shortest Paths

Finds the shortest distance between every pair of vertices in a graph, suitable for sparse graphs with positive or negative edge weights.

Complexity: O(V E log V)
Defined in: <boost/graph/johnson_all_pairs_shortest.hpp>

Example

#include <boost/graph/adjacency_list.hpp>
#include <boost/graph/johnson_all_pairs_shortest.hpp>
#include <iostream>
#include <vector>

struct Node {};
struct Edge { int weight; };

using namespace boost;
using Graph = adjacency_list<vecS, vecS, directedS, Node, Edge>;

int main() {
    Graph g{3};
    add_edge(0, 1, Edge{2}, g);
    add_edge(1, 2, Edge{3}, g);
    add_edge(0, 2, Edge{7}, g);

    auto n = num_vertices(g);
    std::vector<std::vector<int>> D(n, std::vector<int>(n));

    johnson_all_pairs_shortest_paths(g, D,
        get(vertex_index, g), get(&Edge::weight, g), int{0});

    for (std::size_t i = 0; i < n; ++i) {
        for (std::size_t j = 0; j < n; ++j)
            std::cout << D[i][j] << (j + 1 < n ? " " : "");
        std::cout << "\n";
    }
}
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(1) Positional 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);
Direction Parameter Description

IN

VertexAndEdgeListGraph& g1

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.

IN

VertexIndex i_map

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 VertexIndex 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.

IN

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.

IN

DT zero

The value 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.

(2) Positional 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);
Direction Parameter Description

IN

VertexAndEdgeListGraph& g1

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.

IN

VertexID id1

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 VertexID 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.

IN

Weight w1

The weight or "length" of each edge in the graph. The type Weight 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.

IN

const BinaryPredicate& compare

This function is used to compare distances to determine which vertex is closer to the source vertex. The BinaryPredicate type must be a model of Binary Predicate and have argument types that match the value type of the Weight property map.

IN

const BinaryFunction& combine

This function is used to combine distances to compute the distance of a path. The BinaryFunction 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 Weight 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.

IN

const Infinity& inf

This value is used to initialize the distance for each vertex before the start of the algorithm. The type Infinity must be the value type of the Weight map.

IN

DistanceZero zero

This value is used to initialize the distance for the source vertex before the start of the algorithm. The type DistanceZero must be the value type of the Weight map.

(3) Named parameter version

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);
Direction Parameter Description

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.

IN

params

Named parameters passed via bgl_named_params. The following are accepted:
Direction Named Parameter Description / Default

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 used 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.

Description

This algorithm finds the shortest distance between every pair of vertices in the graph. The distance between each pair of vertices is stored in the distance matrix D. The algorithm returns false if there is a negative weight cycle in the graph and true otherwise.

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.

Returns

Returns true if all shortest paths were successfully computed (no negative weight cycle exists). Returns false if a negative weight cycle is detected in the graph.

Example

See example/johnson-eg.cpp for an example that applies Johnson’s algorithm for all-pairs shortest paths to the example graph from page 568 of the CLR [5].

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