An Optimal Minimum Spanning Tree Algorithm

  • Seth Pettie
  • Vijaya Ramachandran
Conference paper
Part of the Lecture Notes in Computer Science book series (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 Gn,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 Gn,m similar to the edge-exposure martingale for Gn,p.

Keywords

Graph algorithms minimum spanning tree optimal complexity 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Seth Pettie
    • 1
  • Vijaya Ramachandran
    • 1
  1. 1.Department of Computer SciencesThe University of Texas at AustinAustin

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