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International Colloquium on Automata, Languages, and Programming

ICALP 2000: Automata, Languages and Programming pp 49–60Cite as

An Optimal Minimum Spanning Tree Algorithm

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1853))

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 forest 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)=Ω(m) and T(m,n)=O(itm·α(m,n)), where α is a certain natural inverse of Ackermann’s function.

Even under the assumption that T is super-linear, we show that if the input graph is selected from G n,m, our algorithm runs in linear time w.h.p., regardless of n, m, or the permutation of edge weights. The analysis uses a new martingale for G n,m similar to the edge-exposure martingale for G n,p.

Part of this work was supported by Texas Advanced Research Program Grant 003658-0029-1999. Seth Pettie was also supported by an MCD Fellowship.

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Author information

Authors and Affiliations

  1. Department of Computer Sciences, The University of Texas at Austin, Austin, TX, 78712

    Seth Pettie & Vijaya Ramachandran

Authors
  1. Seth Pettie
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  2. Vijaya Ramachandran
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Editor information

Editors and Affiliations

  1. Department of Computer Sciences, University of Pisa, Corso Italia, 40, 56125, Pisa, Italy

    Ugo Montanari

  2. Center for Computer Sciences, University of Geneva, 24, Rue Général Dufour, 1211, Geneva 4, Switzerland

    José D. P. Rolim

  3. Department of Computer Sciences, ETH Zurich, 8092, Zurich, Switzerland

    Emo Welzl

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Pettie, S., Ramachandran, V. (2000). An Optimal Minimum Spanning Tree Algorithm. In: Montanari, U., Rolim, J.D.P., Welzl, E. (eds) Automata, Languages and Programming. ICALP 2000. Lecture Notes in Computer Science, vol 1853. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45022-X_6

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  • DOI: https://doi.org/10.1007/3-540-45022-X_6

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67715-4

  • Online ISBN: 978-3-540-45022-1

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