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As a result, disjoint-set forests are both asymptotically optimal and practically efficient. Disjoint-set data structures play a key role in Kruskal's algorithm for finding the minimum spanning tree of a graph. The importance of minimum spanning trees means that disjoint-set data structures support a wide variety of algorithms.
The pseudocode below determines the lowest common ancestor of each pair in P, given the root r of a tree in which the children of node n are in the set n.children. For this offline algorithm, the set P must be specified in advance. It uses the MakeSet, Find, and Union functions of a disjoint-set data structure.
Kruskal's algorithm [1] finds a minimum spanning forest of an undirected edge-weighted graph.If the graph is connected, it finds a minimum spanning tree.It is a greedy algorithm that in each step adds to the forest the lowest-weight edge that will not form a cycle. [2]
Two disjoint sets. In set theory in mathematics and formal logic, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set. [1] For example, {1, 2, 3} and {4, 5, 6} are disjoint sets, while {1, 2, 3} and {3, 4, 5} are not disjoint. A collection of two ...
The problem of determining whether two vertices in a graph are connected can be solved efficiently using a search algorithm, such as breadth-first search. More generally, it is easy to determine computationally whether a graph is connected (for example, by using a disjoint-set data structure), or to
The set system consists of pairwise disjoint sets , …, with sizes ,,, …, respectively, as well as two additional disjoint sets ,, each of which contains half of the elements from each . On this input, the greedy algorithm takes the sets S k , … , S 1 {\displaystyle S_{k},\ldots ,S_{1}} , in that order, while the optimal solution consists ...
The basic form of the Bron–Kerbosch algorithm is a recursive backtracking algorithm that searches for all maximal cliques in a given graph G.More generally, given three disjoint sets of vertices R, P, and X, it finds the maximal cliques that include all of the vertices in R, some of the vertices in P, and none of the vertices in X.
2.1 Intersecting and disjoint sets. 3 Algebraic properties. 4 Arbitrary intersections. 5 Nullary intersection. ... For example, the sets {,} and {,} are disjoint ...