# A minimum spanning tree algorithm with inverse-Ackermann type complexity

@article{Chazelle2000AMS,
title={A minimum spanning tree algorithm with inverse-Ackermann type complexity},
author={Bernard Chazelle},
journal={J. ACM},
year={2000},
volume={47},
pages={1028-1047}
}
• B. Chazelle
• Published 1 November 2000
• Computer Science
• J. ACM
A deterministic algorithm for computing a minimum spanning tree of a connected graph is presented. Its running time is <italic>0</italic>(<italic>m</italic> α(<italic>m, n</italic>)), where α is the classical functional inverse of Ackermann's function and <italic>n</italic> (respectively, <italic>m</italic>) is the number of vertices (respectively, edges). The algorithm is comparison-based : it uses pointers, not arrays, and it makes no numeric assumptions on the edge costs.
341 Citations

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## References

SHOWING 1-10 OF 23 REFERENCES
Finding Minimum Spanning Trees
• Mathematics, Computer Science
SIAM J. Comput.
• 1976
This paper studies methods for finding minimum spanning trees in graphs and results include relationships with other problems which might lead general lower bound for the complexity of the minimum spanning tree problem.
Linear Verification for Spanning Trees
• J. Komlos
• Mathematics, Computer Science
FOCS
• 1984
An algorithm which finds the maximum value of every one of the given paths, and which uses only 5n + n log [(|A|+n)/n] comparisons, leads to a spanning tree verification algorithm using O(n+e) comparisons in a graph with n vertices and e edges.
Linear verification for spanning trees
• J. Komlos
• Mathematics, Computer Science
Comb.
• 1985
An algorithm which finds the maximum value of every one of the given paths, and which uses only O(n+e) comparisons in a graph with n vertices and e edges is described.
Verification and Sensitivity Analysis of Minimum Spanning Trees in Linear Time
• Computer Science
SIAM J. Comput.
• 1992
This paper describes a linear-time algorithm for verifying a minimum spanning tree and combines the result of Komlos with a preprocessing and table look-up method for small subproblems and with a previously known almost-linear- time algorithm.
A randomized linear-time algorithm to find minimum spanning trees
• Computer Science
JACM
• 1995
A randomized linear-time algorithm to find a minimum spanning tree in a connected graph with edge weights is presented, a unit-cost random-access machine with the restriction that the only operations allowed on edge weights are binary comparisons.
Efficient algorithms for finding minimum spanning trees in undirected and directed graphs
• Computer Science
Comb.
• 1986
This paper uses F-heaps to obtain fast algorithms for finding minimum spanning trees in undirected and directed graphs and can be extended to allow a degree constraint at one vertex.
Trans-dichotomous algorithms for minimum spanning trees and shortest paths
• Computer Science
Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science
• 1990
The fusion tree method is extended to develop a linear-time algorithm for the minimum spanning tree problem and an O(m+n log n/log log n) implementation of Dijkstra's shortest-path algorithm for a
On the History of the Minimum Spanning Tree Problem
• Computer Science
Annals of the History of Computing
• 1985
There are several apparently independent sources and algorithmic solutions of the minimum spanning tree problem and their motivations, and they have appeared in Czechoslovakia, France, and Poland, going back to the beginning of this century.
The complexity of computing partial sums off-line
• Mathematics
Int. J. Comput. Geom. Appl.
• 1991
Given an array A with n entries in an additive semigroup, and m intervals of the form Ii=[i,j], where 0<i<j≤n, we show that the computation of A[i]+⋯+A[j] for all Ii, requires Ω(n+mα(m,n)) semigroup