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# minimum_spanning_tree(+Graph, +DistanceArg, -Tree, -TreeWeight)

Computes a minimum spanning tree and its weight

Graph
  a graph structure
DistanceArg
  which argument of EdgeData to use as distance: integer
Tree
  a list of e/3 edge structures
TreeWeight
  sum of the tree's edge weights: number

## Description

Computes a minimum spanning tree for the given graph. A minimum spanning tree is a smallest subset of the graph's edges that still connects all the graph's nodes. Such a tree is not unique and of course exists only if the original graph is itself connected. However, all minimum spanning trees will have the same cost.

The computed tree is returned in Tree, which is simply a list of the edges that form the tree. The TreeWeight is the total length of the tree's edges, according to DistanceArg.

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, and the numeric type (integer, float, etc) must be the same in all edges.

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

The direction of the graph's edges is ignored by this predicate.

The implementation uses Kruskal's algorithm which has a complexity of O(Nedges*log(Nedges)).

### Modes and Determinism

- minimum_spanning_tree(+, +, -, -) is semidet

### Fail Conditions

No spanning tree exists, i.e. the graph is not connected.

## Examples

        ?- sample_graph(G), minimum_spanning_tree(G, 0, T, W).
        T = [e(2, 10, 1), e(4, 8, 1), e(9, 2, 1), e(7, 3, 2), ...]
        W = 16

## See Also

minimum_spanning_forest / 5