# `successive_shortest_path_nonnegative_weights`

    // named parameter version
    template <class Graph, class P, class T, class R>
    void successive_shortest_path_nonnegative_weights(
            Graph &g,
            typename graph_traits<Graph>::vertex_descriptor s,
            typename graph_traits<Graph>::vertex_descriptor t,
            const bgl_named_params<P, T, R> & params  = all defaults)

    // non-named parameter version
    template <class Graph, class Capacity, class ResidualCapacity, class Reversed, class Pred, class Weight, class Distance, class Distance2, class VertexIndex>
    void successive_shortest_path_nonnegative_weights(
            const Graph & g,
            typename graph_traits<Graph>::vertex_descriptor s,
            typename graph_traits<Graph>::vertex_descriptor t,
            Capacity capacity,
            ResidualCapacity residual_capacity,
            Weight weight,
            Reversed rev,
            VertexIndex index,
            Pred pred,
            Distance distance,
            Distance2 distance_prev)

The `successive_shortest_path_nonnegative_weights()` function calculates the minimum cost maximum flow of a network. See Section Network Flow Algorithms for a description of maximum flow. The function calculates the flow values f(u,v) for all (u,v) in E, which are returned in the form of the residual capacity r(u,v) = c(u,v) - f(u,v).

There are several special requirements on the input graph and property map parameters for this algorithm. First, the directed graph G=(V,E) that represents the network must be augmented to include the reverse edge for every edge in E. That is, the input graph should be Gin = (V,{E U ET}). The `ReverseEdgeMap` argument `rev` must map each edge in the original graph to its reverse edge, that is (u,v) -> (v,u) for all (u,v) in E. The `CapacityEdgeMap` argument `cap` must map each edge in E to a positive number, and each edge in ET to 0. The `WeightMap` has to map each edge from E to nonnegative number, and each edge from ET to -weight of its reversed edge.

The algorithm is described in Network Flows.

This algorithm starts with empty flow and in each round augments the shortest path (in terms of weight) in the residual graph.

In order to find the cost of the result flow use: `find_flow_cost()`.

### Where Defined

`boost/graph/successive_shortest_path_nonnegative_weights.hpp`

### Parameters

IN: `Graph& g`

> A directed graph. The graph's type must be a model of VertexListGraph and IncidenceGraph For each edge (u,v) in the graph, the reverse edge (v,u) must also be in the graph.

IN: `vertex_descriptor s`

> The source vertex for the flow network graph.

IN: `vertex_descriptor t`

> The sink vertex for the flow network graph.

### Named Parameters

IN: `capacity_map(CapacityEdgeMap cap)`

> The edge capacity property map. The type must be a model of a constant Lvalue Property Map. The key type of the map must be the graph's edge descriptor type.
> Default: `get(edge_capacity, g)`

OUT: `residual_capacity_map(ResidualCapacityEdgeMap res)`

> This maps edges to their residual capacity. The type must be a model of a mutable Lvalue Property Map. The key type of the map must be the graph's edge descriptor type.
> Default: `get(edge_residual_capacity, g)`

IN: `reverse_edge_map(ReverseEdgeMap rev)`

> An edge property map that maps every edge (u,v) in the graph to the reverse edge (v,u). The map must be a model of constant Lvalue Property Map. The key type of the map must be the graph's edge descriptor type.
> Default: `get(edge_reverse, g)`

IN: `weight_map(WeightMap w_map)`

> The weight or ``cost'' of each edge in the graph. The weights must all be non-negative, and the algorithm will throw a `negative_edge` exception if one of the edges is negative. 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 for this map must be the same as the value type of the distance map.
> Default: `get(edge_weight, g)`

UTIL: `predecessor_map(PredEdgeMap pred)`

> Use by the algorithm to store augmenting paths. The map must be a model of mutable Lvalue Property Map. The key type must be the graph's vertex descriptor type and the value type must be the graph's edge descriptor type.
> Default: an `iterator_property_map` created from a `std::vector` of edge descriptors of size `num_vertices(g)` and using the `i_map` for the index map.

UTIL: `distance_map(DistanceMap d_map)`

> The shortest path weight from the source vertex `s` to each vertex in the graph `g` is recorded in this property map. The shortest path weight is the sum of the edge weights along the shortest path. The type `DistanceMap` must be a model of Read/Write Property Map. The vertex descriptor type of the graph needs to be usable as the key type of the distance map. Default: `iterator_property_map` created from a `std::vector` of the `WeightMap`'s value type of size `num_vertices(g)` and using the `i_map` for the index map.

UTIL: `distance_map2(DistanceMap2 d_map2)`

> The shortest path computation in iteration nr k uses distances computed in iteration k. The type `DistanceMap2` must be a model of Read/Write Property Map. The vertex descriptor type of the graph needs to be usable as the key type of the distance map. Default: `iterator_property_map` created from a `std::vector` of the `WeightMap`'s value type of size `num_vertices(g)` and using the `i_map` for the index map.

IN: `vertex_index_map(VertexIndexMap i_map)`

> Maps each vertex of the graph to a unique integer in the range `[0, num_vertices(g))`. This property map is only needed if the default for the distance or distance2 or predecessor map is used. The vertex index map must be a model of Readable Property Map. The key type of the map must be the graph's vertex descriptor type.
> 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.

### Complexity

In the integer capacity case, if U is the value of the max flow, then the complexity is O(U * (|E| + |V|*log|V|)), where O(|E| + |V|*log|V|) is the complexity of the dijkstra algorithm and U is upper bound on number of iteration. In many real world cases number of iterations is much smaller than U.

### Example

The program in `example/successive_shortest_path_nonnegative_weights_example.cpp`.

### See Also

`cycle_canceling()`
`find_flow_cost()`.

Copyright © 2013 | Piotr Wygocki, University of Warsaw (wygos at mimuw.edu.pl)