Article

Combinatorica

, Volume 6, Issue 2, pp 109-122

First online:

Efficient algorithms for finding minimum spanning trees in undirected and directed graphs

  • Harold N. GabowAffiliated withUniversity of Colorado
  • , Zvi GalilAffiliated withColumbia UniversityTel Aviv University
  • , Thomas SpencerAffiliated withRensselaer Polytechnic Inst.
  • , Robert E. TarjanAffiliated withAT&T Bell Laboratories

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Abstract

Recently, Fredman and Tarjan invented a new, especially efficient form of heap (priority queue). Their data structure, theFibonacci heap (or F-heap) supports arbitrary deletion inO(logn) amortized time and other heap operations inO(1) amortized time. In this paper we use F-heaps to obtain fast algorithms for finding minimum spanning trees in undirected and directed graphs. For an undirected graph containingn vertices andm edges, our minimum spanning tree algorithm runs inO(m logβ (m, n)) time, improved fromO((m, n)) time, whereβ(m, n)=min {i|log(i) nm/n}. Our minimum spanning tree algorithm for directed graphs runs inO(n logn + m) time, improved fromO(n log n +m log log log(m/n+2) n). Both algorithms can be extended to allow a degree constraint at one vertex.

AMS subject classification (1980)

68 B 15 68 C 05