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This updating is an important part of the disjoint-set forest's amortized performance guarantee. There are several algorithms for Find that achieve the asymptotically optimal time complexity. One family of algorithms, known as path compression, makes every node between the query node and the root point to the root. Path compression can be ...
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.
Dominating set, a.k.a. domination number [3]: GT2 NP-complete special cases include the edge dominating set problem, i.e., the dominating set problem in line graphs. NP-complete variants include the connected dominating set problem and the maximum leaf spanning tree problem. [3]: ND2 Feedback vertex set [2] [3]: GT7
Set packing is a classical NP-complete problem in computational complexity theory and combinatorics, and was one of Karp's 21 NP-complete problems. Suppose one has a finite set S and a list of subsets of S. Then, the set packing problem asks if some k subsets in the list are pairwise disjoint (in other words, no two of them share an element).
The problem of finding two disjoint paths of minimum weight can be seen as a special case of a minimum cost flow problem, where in this case there are two units of "flow" and nodes have unit "capacity". Suurballe's algorithm, also, can be seen as a special case of a minimum cost flow algorithm that repeatedly pushes the maximum possible amount ...
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 ...
Each vertex of the graph is a part of a path, including vertex D, which is a part of a path with length 0. The set of such paths is a path cover. A path cover may also refer to a vertex-disjoint path cover, i.e., a set of paths such that every vertex v ∈ V belongs to exactly one path. [2]
The vertex-connectivity statement of Menger's theorem is as follows: . Let G be a finite undirected graph and x and y two nonadjacent vertices. Then the size of the minimum vertex cut for x and y (the minimum number of vertices, distinct from x and y, whose removal disconnects x and y) is equal to the maximum number of pairwise internally disjoint paths from x to y.