
single_pair_all_short_paths_as_graph(+Graph, +DistanceArg, +SourceNode, +SinkNode, +Tolerance, -Length, -SubGraph)

   Computes all shortest paths from source to sink in form of a subgraph

Arguments
   Graph               a graph structure
   DistanceArg         which argument of EdgeData to use as distance (integer)
   SourceNode          source node number (integer)
   SinkNode            sink node number (integer)
   Tolerance           tolerable deviation from minimal length (non-negative number)
   Length              a number (minimum path length)
   SubGraph            a graph structure

Type
   library(graph_algorithms)

Description

    Computes all shortest paths from source node SourceNode to sink
    node SinkNode.  The result is returned in the form of a sub-graph
    of the input graph, which contains all relevant edges.
    If there is no path, the predicate fails.

    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. Important: the distance
    information in EdgeData must be a non-negative number.

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

    Tolerance should be zero in order to find only the shortest paths. 
    If Tolerance is greater than zero, then all paths that are within
    this tolerance of the shortest path length will be found.

    The result is returned in the form of SubGraph, which is a
    sub-graph of the input Graph, containing the same nodes, but only
    those edges that are needed to construct the shortest paths for
    the given parameters.  SubGraph does not inherit the nodename
    information from Graph, this can be set explicitly if required.

    In addition, the Length of the shortest path from source to sink
    is returned.

Properties of the resulting SubGraph

    To generate an actual path from the resulting SubGraph, start from the
    sink node J, select one of its incoming edges (graph_get_incoming_edges/3)
    to find a predecessor node, and continue this process until the SourceNode
    is reached. Depending on the parameters, the following 3 cases can occur:
    
    Tolerance is zero, and Graph did not contain zero-length edges: in this
    case, SubGraph is cycle-free and shortest paths can be found by simply
    selecting arbitrary incoming edges until SourceNode is reached.
    Tolerance is zero, and Graph did contain zero-length edges: in this case,
    SubGraph may contain (zero-length) cycles which one may want to exclude
    when constructing paths.
    Tolerance is nonzero:  in this case, SubGraph may contain
    cycles (of maximum length Tolerance).  Moreover, it may be
    possible to use the edges in SubGraph to construct cycle-free paths
    whose total length is greater than the shortest path length plus
    the tolerance.  These may need to be excluded explicitly.
    
    

Modes and Determinism
   single_pair_all_short_paths_as_graph(+, +, +, +, +, -, -) is det

Examples
   
    ?- sample_graph(G),
       single_pair_all_short_paths_as_graph(G, 0, 1, 5, 0, L, E).
    G = graph(13, []([e(1, 6, 1), e(1, 2, 1), e(1, 7, 1)], [], ...)
    L = 2
    SG = graph(13, []([e(1, 6, 1), e(1, 7, 1)], [], ...)
    

See Also
   possible_path / 7, shortest_paths / 4, single_pair_shortest_path / 5, all_short_paths_as_edges / 6, all_short_paths_as_graph / 6, single_pair_short_path / 6, graph_get_incoming_edges / 3, graph_set_nodenames / 2
