# `union_find`

This library implements a union-find data structure. This structure tracks a set of elements partitioned into a number of disjoint (non-overlapping) subsets. It provides fast operations to add new sets, to merge existing sets, and to determine whether elements are in the same set. This implementation of the union-find algorithm provides the following features:

- Path compression: Path compression flattens the structure of the tree by making every node point to the root whenever a find predicate is used on it.

- Union by rank: Union predicates always attach the shorter tree to the root of the taller tree. Thus, the resulting tree is no taller than the original unless they were of equal height, in which case the resulting tree is taller by one node.

For a general and extended discussion on this data structure, see e.g.

https://en.wikipedia.org/wiki/Disjoint-set_data_structure

## API documentation

Open the ../../apis/library_index.html#union-find link in a web browser.

## Loading

To load all entities in this library, load the `loader.lgt` file:

    | ?- logtalk_load(union_find(loader)).

## Testing

To test this library predicates, load the `tester.lgt` file:

    | ?- logtalk_load(union_find(tester)).

## Usage

An usage example is Kruskal’s algorithm, a minimum-spanning-tree algorithm which finds an edge of the least possible weight that connects any two trees in the forest. It is a greedy algorithm in graph theory as it finds a minimum spanning tree for a connected weighted graph by adding increasing cost arcs at each step.

    :- object(kruskal).

        :- public(kruskal/2).

        :- uses(union_find, [
            new/2, find/4, union/4
        ]).

        kruskal(g(Vertices-Edges), g(Vertices-Tree)) :-
            new(Vertices, UnionFind),
            keysort(Edges, Sorted),
            kruskal(UnionFind, Sorted, Tree).

        kruskal(_, [], []).
        kruskal(UnionFind0, [Edge| Edges], [Edge| Tree]) :-
            Edge = _-(Vertex1, Vertex2),
            find(UnionFind0, Vertex1, Root1, UnionFind1),
            find(UnionFind1, Vertex2, Root2, UnionFind2),
            Root1 \== Root2,
            !,
            union(UnionFind2, Vertex1, Vertex2, UnionFind3),
            kruskal(UnionFind3, Edges, Tree).
        kruskal(UnionFind, [_| Edges], Tree) :-
            kruskal(UnionFind, Edges, Tree).

    :- end_object.

Sample query:

    | ?- kruskal::kruskal(g([a,b,c,d,e,f,g]-[7-(a,b), 5-(a,d), 8-(b,c), 7-(b,e), 9-(b,d), 5-(c,e), 15-(d,e), 6-(d,f), 8-(e,f), 9-(e,g), 11-(f,g)]), Tree).

    Tree = g([a,b,c,d,e,f,g]-[5-(a,d),5-(c,e),6-(d,f),7-(a,b),7-(b,e),9-(e,g)])
    yes