An optimal minimum spanning tree algorithm

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An optimal minimum spanning tree algorithm (2000)

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by Seth Pettie , Vijaya Ramachandran

Venue:	J. ACM
Citations:	58 - 11 self

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@ARTICLE{Pettie00anoptimal,
    author = {Seth Pettie and Vijaya Ramachandran},
    title = {An optimal minimum spanning tree algorithm},
    journal = {J. ACM},
    year = {2000},
    volume = {49},
    pages = {49--60}
}

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Abstract

Abstract. We establish that the algorithmic complexity of the minimum spanning tree problem is equal to its decision-tree complexity. Specifically, we present a deterministic algorithm to find a minimum spanning tree of a graph with n vertices and m edges that runs in time O(T ∗ (m, n)) where T ∗ is the minimum number of edge-weight comparisons needed to determine the solution. The algorithm is quite simple and can be implemented on a pointer machine. Although our time bound is optimal, the exact function describing it is not known at present. The current best bounds known for T ∗ are T ∗ (m, n) = �(m) and T ∗ (m, n) = O(m · α(m, n)), where α is a certain natural inverse of Ackermann’s function. Even under the assumption that T ∗ is superlinear, we show that if the input graph is selected from Gn,m, our algorithm runs in linear time with high probability, regardless of n, m, or the permutation of edge weights. The analysis uses a new martingale for Gn,m similar to the edge-exposure martingale for Gn,p.

Keyphrases

optimal minimum spanning tree algorithm new martingale minimum spanning tree problem edge-exposure martingale deterministic algorithm minimum number certain natural inverse linear time pointer machine time bound edge weight minimum spanning tree ackermann function algorithmic complexity high probability exact function input graph decision-tree complexity edge-weight comparison

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