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stoer_wagner_min_cut

Determines a min-cut and the min-cut weight of a connected, undirected graph.

Complexity: O(V * E + V2 log V)
Defined in: <boost/graph/stoer_wagner_min_cut.hpp>

Example

#include <boost/graph/adjacency_list.hpp>
#include <boost/graph/stoer_wagner_min_cut.hpp>
#include <boost/property_map/property_map.hpp>
#include <iostream>

struct EdgeProps { int weight; };

using namespace boost;

using Graph = adjacency_list<vecS, vecS, undirectedS,
    no_property, EdgeProps>;

int main() {
    Graph g{4};
    add_edge(0, 1, EdgeProps{2}, g);
    add_edge(0, 2, EdgeProps{3}, g);
    add_edge(1, 2, EdgeProps{3}, g);
    add_edge(1, 3, EdgeProps{2}, g);
    add_edge(2, 3, EdgeProps{4}, g);

    auto weight_map = get(&EdgeProps::weight, g);
    int cut = stoer_wagner_min_cut(g, weight_map);
    std::cout << "Stoer-Wagner min cut: " << cut << "\n";
}
Stoer-Wagner min cut: 5

(1) Named parameter version

A min-cut of a weighted graph having min-cut weight 4
Figure 1. A min-cut of a weighted graph having min-cut weight 4
template <class UndirectedGraph, class WeightMap,
          class P, class T, class R>
weight_type stoer_wagner_min_cut(
    const UndirectedGraph& g,
    WeightMap weights,
    const bgl_named_params<P, T, R>& params = all defaults);
Direction Parameter Description

IN

const UndirectedGraph& g

A connected, undirected graph. The graph type must be a model of Vertex List Graph and Incidence Graph.

IN

WeightMap weights

The weight or length of each edge in the graph. The WeightMap type must be a model of Readable Property Map and its value type must be Less Than Comparable and summable. The key type of this map needs to be the graph’s edge descriptor type.
Direction Named Parameter Description / Default

OUT

parity_map(ParityMap parities)

The algorithm computes a min-cut, which divides the set of vertices into two, non-empty sets. The stoer_wagner_min_cut function records which of the two sets that each vertex belongs to by setting the parity to true (representing one set) or false (for the other). ParityMap must be a model of a Writable Property Map and its value type should be a bool type. The key type must be the graph’s vertex descriptor type.
Default: boost::dummy_property_map

IN

vertex_index_map(VertexIndexMap vertexIndices)

This maps each vertex to an integer in the range [0, num_vertices(g)). This is only necessary if the default is used for the assignment, index-in-heap, or distance maps. VertexIndexMap must be a model of Readable Property Map. The value type of the map must be an integer type. The key type must be the graph’s vertex descriptor type.
Default: get(boost::vertex_index, g)

UTIL

assignment_map(AssignmentMap assignments)

AssignmentMap must be a model of Read/Write Property Map. The key and value types must be the graph’s vertex descriptor type.
Default: A boost::iterator_property_map using a std::vector of num_vertices(g) vertex descriptors and vertexIndices for the index map.

UTIL

max_priority_queue(MaxPriorityQueue& pq)

MaxPriorityQueue must be a model of Keyed Updatable Queue and a max-Updatable Priority Queue. The value type must be the graph’s vertex descriptor and the key type must be the weight type.
Default: A boost::d_ary_heap_indirect using a default index-in-heap and distance map.

UTIL

index_in_heap_map(IndexInHeapMap indicesInHeap)

This parameter only has an effect when the default max-priority queue is used.
IndexInHeapMap must be a model of Read/Write Property Map. The key type must be the graph’s vertex descriptor type. The value type must be a size type (typename std::vector<vertex_descriptor>::size_type).
Default: A boost::iterator_property_map using a std::vector of num_vertices(g) size type objects and vertexIndices for the index map.

UTIL

distance_map(DistanceMap wAs)

This parameter only has an effect when the default max-priority queue is used.
DistanceMap must be a model of Read/Write Property Map. The key type must be the graph’s vertex descriptor type. The value type must be the weight type (typename boost::property_traits<WeightMap>::value_type).
Default: A boost::iterator_property_map using a std::vector of num_vertices(g) weight type objects and vertexIndices for the index map.

(2) Positional version (explicit index map)

template <class UndirectedGraph, class WeightMap, class ParityMap,
          class VertexAssignmentMap, class KeyedUpdatablePriorityQueue,
          class IndexMap>
typename property_traits<WeightMap>::value_type
stoer_wagner_min_cut(
    const UndirectedGraph& g, WeightMap weights,
    ParityMap parities, VertexAssignmentMap assignments,
    KeyedUpdatablePriorityQueue& pq, IndexMap index_map);

Explicit positional form with every working map supplied by the caller. Prefer overload (1) unless you have a reason to pre-allocate the work maps and priority queue.

(3) Positional version (default index map)

template <class UndirectedGraph, class WeightMap, class ParityMap,
          class VertexAssignmentMap, class KeyedUpdatablePriorityQueue>
typename property_traits<WeightMap>::value_type
stoer_wagner_min_cut(
    const UndirectedGraph& g, WeightMap weights,
    ParityMap parities, VertexAssignmentMap assignments,
    KeyedUpdatablePriorityQueue& pq);

Lives in namespace boost::graph. Same as (2) with index_map = get(vertex_index, g); requires the graph to have an interior vertex_index_t property.

Description

A cut of a graph G is a partition of the vertices into two, non-empty sets. The weight of such a partition is the number of edges between the two sets if G is unweighted, or the sum of the weights of all edges between the two sets if G is weighted. A min-cut is a cut having the least weight.

Sometimes a graph has multiple min-cuts, but all have the same weight. The stoer_wagner_min_cut function implements the Stoer-Wagner algorithm and determines exactly one of the min-cuts as well as its weight.

Returns

The weight of the min-cut.

Throws

bad_graph

If num_vertices(g) is less than 2.

std::invalid_argument

If a max-priority queue is given as an argument and it is not empty.

Example

The file examples/stoer_wagner.cpp contains an example of calculating a min-cut of a weighted, undirected graph and its min-cut weight.

References

  • [74] Mehlhorn, Kurt and Christian Uhrig (1995). The minimum cut algorithm of Stoer and Wagner.

  • [75] Stoer, Mechthild and Frank Wagner (1997). A simple min-cut algorithm. Journal of the ACM 44 (4), 585-591.

  • [76] Zwick, Uri (2008). Global minimum cuts.

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