# An optimal minimum spanning tree algorithm

@article{Pettie2002AnOM,
title={An optimal minimum spanning tree algorithm},
author={Seth Pettie and Vijaya Ramachandran},
journal={J. ACM},
year={2002},
volume={49},
pages={16-34}
}
• Published 2002
• Mathematics, Computer Science
• J. ACM
We establish that the algorithmic complexity of the minimumspanning tree problem is equal to its decision-tree complexity.Specifically, we present a deterministic algorithm to find aminimum spanning tree of a graph with <i>n</i> vertices and<i>m</i> edges that runs in time<i>O</i>(<i>T</i><sup>*</sup>(<i>m,n</i>)) where<i>T</i><sup>*</sup> is the minimum number of edge-weightcomparisons needed to determine the solution. The algorithm isquite simple and can be implemented on a pointer machine… Expand
269 Citations

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#### References

SHOWING 1-10 OF 69 REFERENCES
An Optimal Minimum Spanning Tree Algorithm
• Mathematics, Computer Science
• ICALP
• 2000
A deterministic algorithm to find a minimum spanning forest of a graph with n vertices and m edges that runs in time O(Τ*(m, n) where Τ* is the minimum number of edge-weight comparisons needed to determine the solution, which is quite simple and can be implemented on a pointer machine. Expand
Computing shortest paths with comparisons and additions
• Mathematics, Computer Science
• SODA '02
• 2002
An undirected all-pairs shortest paths (APSP) algorithm which runs on a pointer machine in time in time while making comparisons and additions, where m and n are the number of edges and vertices, respectively, and α(<i>m, n</i) is Tarjan's inverse-Ackermann function. Expand
Applications of Path Compression on Balanced Trees
• R. Tarjan
• Mathematics, Computer Science
• JACM
• 1979
A method for computing functions defined on paths in trees based on tree manipulation techniques first used for efficiently representing equivalence relations, which has an almost-linear running time and is useful for solving certain kinds of pathfinding problems on reducible graphs. Expand
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. Expand
A minimum spanning tree algorithm with inverse-Ackermann type complexity
A deterministic algorithm for computing a minimum spanning tree of a connected graph that uses pointers, not arrays, and it makes no numeric assumptions on the edge costs. Expand
Minimizing randomness in minimum spanning tree, parallel connectivity, and set maxima algorithms
• Mathematics, Computer Science
• SODA '02
• 2002
This work considers the problem of selection and proposes new algorithms for these problems which preserve optimality while saving an exponential number of random bits, in the case of computing minimum spanning trees and MST/SSSP sensitivity analysis. Expand
Concurrent threads and optimal parallel minimum spanning trees algorithm
• Computer Science
• JACM
• 2001
This paper resolves a long-standing open problem on whether the concurrent write capability of parallel random access machine (PRAM) is essential for solving fundamental graph problems like connectedExpand
Efficient algorithms for finding minimum spanning trees in undirected and directed graphs
• Mathematics, 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. Expand
Verification and Sensitivity Analysis of Minimum Spanning Trees in Linear Time
• Mathematics, 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. Expand
A faster deterministic algorithm for minimum spanning trees
• B. Chazelle
• Mathematics, Computer Science
• Proceedings 38th Annual Symposium on Foundations of Computer Science
• 1997
A deterministic algorithm for computing a minimum spanning tree of a connected graph is presented, which improves on the previous, ten-year old bound of (roughly) O(m log log* m). Expand