
all_short_paths_as_edges(+Graph, +DistanceArg, +SourceNode, +Tolerance, -Lengths, -Predecessors)

   Computes all shortest paths from a single source to every reachable node

Arguments
   Graph               a graph structure
   DistanceArg         which argument of EdgeData to use as distance (integer)
   SourceNode          source node number (integer)
   Tolerance           tolerable deviation from minimal length (non-negative number)
   Lengths             array of numbers (minimum path lengths)
   Predecessors        array of lists of e/3 edge structures

Type
   library(graph_algorithms)

Description

    Computes all shortest paths from the single source node SourceNode
    to every sink node which is reachable from it. The result is returned
    in the form of the Predecessors array which contains all relevant edges.

    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.

    SourceNode is the common starting point for the computed paths.

    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 two arrays, whose indices range
    over the possible sink nodes.  The Lengths array indicates the length
    of a shortest path from SourceNode to the corresponding sink node.
    The Predecessors array is an array of edge lists, each list containing
    the alternative edges that are part of a shortest path from SourceNode
    to the corresponding sink node.

    If there is no path from SourceNode to a sink node J, then both
    Lengths[J] and Predecessors[J] are uninstantiated. Otherwise,
    Lengths[J] contains the length of a shortest path from SourceNode to J.
    Predecessors[J] is contains a list of alternative edges that lead from
    some predecessor node to J in a shortest path from SourceNode to J.
    Predecessors[SourceNode] is always the empty list [].

Assembling Actual Paths

    To generate an actual path from the Predecessors array, start from the
    sink node J, select one of the alternative edges in Predecessors[J]
    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.
    
    The possible_path/7 predicate implements this path construction and
    does the necesssary checks to exclude cycles and overly long paths.
    

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

Examples
   
    ?- sample_graph(G), all_short_paths_as_edges(G, 0, 1, 0, L, E).
    L = [](0, 1, 2, 3, 2, 1, 1, _326, _327, 2, 3, 3, 3)
    E = []([], [e(1, 2, 1)], [e(7, 3, 1)], [e(5, 4, 1)],
	   [e(7, 5, 1), e(6, 5, 1)], [e(1, 6, 1)], [e(1, 7, 1)],
	   _342, _343, [e(7, 10, 1)], [e(10, 11, 1)], [e(10, 12, 1)],
	   [e(10, 13, 1)])
    

See Also
   possible_path / 7, shortest_paths / 4, single_pair_shortest_path / 5, all_short_paths_as_graph / 6, single_pair_short_path / 6, single_pair_all_short_paths_as_graph / 7
