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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.
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]
The following example shows how Suurballe's algorithm finds the shortest pair of disjoint paths from A to F. Figure A illustrates a weighted graph G. Figure B calculates the shortest path P 1 from A to F (A–B–D–F). Figure C illustrates the shortest path tree T rooted at A, and the computed distances from A to every vertex (u).
The vertex disjoint version of the above edge-disjoint shortest pair of paths algorithm is obtained by splitting each vertex (except for the source and destination vertices) of the first shortest path in Step 3 of the algorithm, connecting the split vertex pair by a zero weight arc (directed towards the source vertex), and replacing any ...
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.
A minimum path cover consists of one path if and only if there is a Hamiltonian path in G. The Hamiltonian path problem is NP-complete, and hence the minimum path cover problem is NP-hard. However, if the graph is acyclic, the problem is in complexity class P and can therefore be solved in polynomial time by transforming it into a matching ...
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