An optimal minimum spanning tree algorithm

@inproceedings{Pettie2002AnOM,
  title={An optimal minimum spanning tree algorithm},
  author={Seth Pettie and Vijaya Ramachandran},
  booktitle={JACM},
  year={2002}
}
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… 

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References

SHOWING 1-10 OF 58 REFERENCES
Computing shortest paths with comparisons and additions
TLDR
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.
Trans-dichotomous algorithms for minimum spanning trees and shortest paths
  • M. Fredman, D. Willard
  • 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
Applications of Path Compression on Balanced Trees
TLDR
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.
Finding Minimum Spanning Trees
TLDR
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.
A minimum spanning tree algorithm with inverse-Ackermann type complexity
TLDR
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.
Minimizing randomness in minimum spanning tree, parallel connectivity, and set maxima algorithms
TLDR
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.
Fibonacci heaps and their uses in improved network optimization algorithms
TLDR
Using F-heaps, a new data structure for implementing heaps that extends the binomial queues proposed by Vuillemin and studied further by Brown, the improved bound for minimum spanning trees is the most striking.
Concurrent threads and optimal parallel minimum spanning trees algorithm
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 connected
Efficient algorithms for finding minimum spanning trees in undirected and directed graphs
TLDR
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.
Verification and Sensitivity Analysis of Minimum Spanning Trees in Linear Time
TLDR
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.
...
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