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

```@article{Gabow1986EfficientAF,
title={Efficient algorithms for finding minimum spanning trees in undirected and directed graphs},
author={Harold N. Gabow and Zvi Galil and Thomas H. Spencer and Robert E. Tarjan},
journal={Combinatorica},
year={1986},
volume={6},
pages={109-122}
}```
• Published 2 January 1986
• Computer Science
• Combinatorica
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…
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