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# minimum_spanning_forest(+Graph, +DistanceArg, -Forest, -ForestSize, -ForestWeight)

Computes a minimum spanning forest, its size and weight
*Graph*
- a graph structure
*DistanceArg*
- which argument of EdgeData to use as distance: integer
*Forest*
- a list of e/3 edge structures
*ForestSize*
- the number of edges in the Forest list
*ForestWeight*
- sum of the forest's edge weights: number

## Description

Computes a minimum spanning forest for the given graph. A minimum
spanning forest is a smallest subset of the graph's edges that still
connects all the graph's connected components. Such a forest is not
unique, but all minimum spanning forests will have the same cost.
As opposed to a minimum spanning tree, a forest exists also if the
original graph is not connnected. The forest will have the same
number of connected components as the original graph.

The computed forest is returned in Forest, which is simply a list of
the edges that form the forest. The ForestSize is the number of
edges that constitute the forest. The ForestWeight is the total
length of the forest'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_forest(+, +, -, -, -) is det

## Examples

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

## See Also

minimum_spanning_tree / 4