Efficient algorithms for finding minimum spanning trees in undirected and directed graphs. Gabow, H., Galil, Z., Spencer, T., & Tarjan, R. *Combinatorica*, 6(2):109–122, Springer Berlin / Heidelberg, 1986. doi abstract bibtex Recently, Fredman and Tarjan invented a new, especially efficient form of heap (priority queue). Their data structure, the Fibonacci heap (or F-heap) supports arbitrary deletion inO(logn) amortized time and other heap operations in O(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 containing n vertices and m edges, our minimum spanning tree algorithm runs in O(m log β (m, n)) time, improved fromO(m β(m, n)) time, where β(m, n) = min i | log^(i) n łe m/n. Our minimum spanning tree algorithm for directed graphs runs in O(n log n + m) time, improved from O(n log n +m log log log(m/n+2) n). Both algorithms can be extended to allow a degree constraint at one vertex.

@Article{gabow86efficient,
author = {Gabow, Harold and Galil, Zvi and Spencer, Thomas and Tarjan, Robert},
title = {Efficient algorithms for finding minimum spanning trees in undirected and directed graphs},
journal = {Combinatorica},
year = {1986},
volume = {6},
pages = {109--122},
issn = {0209-9683},
abstract = {Recently, Fredman and Tarjan invented a new, especially efficient form of heap (priority queue). Their data structure, the Fibonacci heap (or F-heap) supports arbitrary deletion inO(logn) amortized time and other heap operations in O(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 containing n vertices and m edges, our minimum spanning tree algorithm runs in O(m log \beta (m, n)) time, improved fromO(m \beta(m, n)) time, where \beta(m, n) = min {i | log^{(i)} n \le m/n}. Our minimum spanning tree algorithm for directed graphs runs in O(n log n + m) time, improved from O(n log n +m log log log(m/n+2) n). Both algorithms can be extended to allow a degree constraint at one vertex.},
affiliation = {University of Colorado 80309 Boulder CO USA},
doi = {10.1007/BF02579168},
number = {2},
keywords = {maximum spanning tree, directed graphs},
owner = {Sebastian},
publisher = {Springer Berlin / Heidelberg},
timestamp = {2011.02.02},
}

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