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

  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},
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… 
Bottleneck paths and trees and deterministic graphical
The Gabow and Tarjan problem, i.e., finding a path between a given source and a given target in a weighted directed graph whose largest edge weight is minimized, as well as the Bottleneck spanning tree problem, can be solved deterministically in O(m log∗ n) time.
A Flexible Algorithm for Generating All the Spanning Trees in Undirected Graphs
An algorithm for generating all the spanning trees in undirected graphs that requires O (n+m+ τ n) time where the given graph has n vertices, m edges, and τ spanning trees.
Parallel algorithms for verification and sensitivity analysis of minimum spanning trees
An efficient algorithm for sensitivity analysis of minimum spanning trees which requires O(log n) time and O(max{m, n/sup 2//log n}) processors is presented and has better performance when G is sparse.
Bottleneck Paths and Trees and Deterministic Graphical Games
A slightly improved randomized algorithm for the Bottleneck Path problem and the Bott bottleneck spanning tree problem, observing that in the word-RAM model, both problems can be solved deterministically in O(m) time.
Trans-dichotomous algorithms for minimum spanning trees and shortest paths
  • M. Fredman, D. Willard
  • Computer Science
    Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science
  • 1990
The fusion tree method is extended to develop a linear-time algorithm for the minimum spanning tree problem and an O(m+n log n/log log n) implementation of Dijkstra's shortest-path algorithm for a
Representing all minimum spanning trees with applications to counting and generation
It is shown that for any edge-weighted graph G there is an equivalent graph EG such that the minimum spanning trees of G correspond one-for-one with the spanning Trees of EG, which can be constructed in the time O(m + n log n) given a single minimum spanning tree of G.
Finding the k Most Vital Edges in the Minimum Spanning Tree Problem
Lecture notes on “Analysis of Algorithms”: Directed Minimum Spanning Trees (More complete but still unfinished)
We describe an efficient implementation of Edmonds’ algorithm for finding minimum directed spanning trees in directed graphs. 1 Minimum Directed Spanning Trees Let G = (V,E,w) be a weighted directed


Applications of Path Compression on Balanced Trees
A method for computing functions defined on paths in trees based on tree manipulation techniques first used for efficiently representing equivalence relations, which has an almost-linear running time and is useful for solving certain kinds of pathfinding problems on reducible graphs.
Finding Minimum Spanning Trees
This paper studies methods for finding minimum spanning trees in graphs and results include relationships with other problems which might lead general lower bound for the complexity of the minimum spanning tree problem.
Finding optimum branchings
  • R. Tarjan
  • Mathematics, Computer Science
  • 1977
An implementation of the algorithm which runs in 0(m logn) time if the problem graph has n vertices and m edges is given, and a modification for dense graphs gives a running time of 0(n2).
Efficient Algorithms for a Family of Matroid Intersection Problems
On the History of the Minimum Spanning Tree Problem
There are several apparently independent sources and algorithmic solutions of the minimum spanning tree problem and their motivations, and they have appeared in Czechoslovakia, France, and Poland, going back to the beginning of this century.
A note on finding optimum branchings
Tarjan's algorithm for finding an optimum branching in a directed graph is shown to have two errors, namely an incorrect claim involving branching uniqueness, and an imprecise way of updating edge values in each iteration.
Efficiency of a Good But Not Linear Set Union Algorithm
It is shown that, if t(m, n) is seen as the maximum time reqmred by a sequence of m > n FINDs and n -- 1 intermixed UNIONs, then kima(m), n is shown to be related to a functional inverse of Ackermann's functmn and as very slow-growing.
A simple derivation of Edmonds' algorithm for optimum branchings
A direct combinatorial proof of the correctness of the algorithm for constructing a maximum-weight branching in a weighted directed graph based on linear programming theory is given.
Worst-case Analysis of Set Union Algorithms
It is shown that two one-pass methods proposed by van Leeuwen and van der Weide are asymptotically optimal, whereas several other methods, including one proposed by Rein and advocated by Dijkstra, are slower than the best methods.