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# shortest_paths_bellman_ford(+Graph, +DistanceArg, +SourceNode, -Paths)

Computes one shortest path from a single source to every reachable node (allowing negative distances)

Graph
  a graph structure
DistanceArg
  which argument of EdgeData to use as distance: integer
SourceNode
  source node number
Paths
  array of Length-EdgeList structures

## Description

Computes one shortest path from the single source node SourceNode to every node which is reachable from it. In case of multiple shortest paths with the same length, an arbitrary one is returned.

DistanceArg refers to the graph's EdgeData information that was specified when the graph was constructed. If EdgeData is a simple number, then DistanceArg should be 0 and EdgeData will be taken as the length of the edge. If EdgeData is a compound data structure, DistanceArg should be a number between 1 and the arity of that structure and determines which argument of the EdgeData structure will be interpreted as the edge's length. As opposed to the other shortest path algorithms, the Bellman-Ford algorithm can handle negative edge lengths, however, the implementation has currently no check for negative cycles and will not terminate in that case.

If DistanceArg is given as -1, then any EdgeData is ignored and the length of every edge is assumed to be equal to 1.

SourceNode is the common starting point for the computed paths.

The results are returned as an array ranging over all node numbers. For unreachable nodes the array element is uninstantiated. For reachable nodes, the element contains a Length-EdgeList structure where Length is the length of the shortest path and EdgeList is that path (or one of them) in reverse order, i.e. starting with the edge reaching the target and ending with the edge starting from SourceNode.

### Modes and Determinism

- shortest_paths_bellman_ford(+, +, +, -) is det

## Examples

        ?- sample_graph(G), shortest_paths_bellman_ford(G, 0, 1, P).
        P = [](2 - [e(2, 1, 1), e(1, 2, 1)], 1 - [e(1, 2, 1)], ...)

## See Also

single_pair_shortest_path_bellman_ford / 5, shortest_paths / 4