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# all_short_paths_as_graph(+Graph, +DistanceArg, +SourceNode, +Tolerance, -Lengths, -SubGraph)

Computes all shortest paths from a single source in form of a subgraph
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)
SubGraph
a graph structure

## 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 a sub-graph of the input graph, 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 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, a Lengths array is returned, whose entries indicate the length of a shortest path from SourceNode to the corresponding sink node. If there is no path from SourceNode to a sink node J, then Lengths[J] is uninstantiated.

#### 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:

1. 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.
2. 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.
3. 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

• all_short_paths_as_graph(+, +, +, +, -, -) is det

## Examples

```    ?- sample_graph(G), all_short_paths_as_graph(G, 0, 1, 0, L, E).
G = graph(13, []([e(1, 6, 1), e(1, 2, 1), e(1, 7, 1)], [], ...)
L = [](0, 1, 2, 3, 2, 1, 1, _326, _327, 2, 3, 3, 3)
SG = graph(13, []([e(1, 7, 1), e(1, 6, 1), e(1, 2, 1)], [], ...)
Yes (0.00s cpu)
```