________________________________________________________________________

This file is part of Logtalk <https://logtalk.org/>  
Copyright 2022 José Antonio Riaza Valverde <riazavalverde@gmail.com>  
Copyright 2022 Paulo Moura <pmoura@logtalk.org>  
SPDX-License-Identifier: Apache-2.0

Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at

    http://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
________________________________________________________________________


`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 originals
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 [../../docs/library_index.html#union-find](../../docs/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 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
