Verification and Sensitivity Analysis of Minimum Spanning Trees in Linear Time

Komlos has devised a way to use a linear number of binary comparisons to test whether a given spanning tree of a graph with edge costs is a minimum spanning tree. The total computational work required by his method is much larger than linear, however. This paper describes a linear-time algorithm for verifying a minimum spanning tree. This algorithm combines the result of Komlos with a preprocessing and table look-up method for small subproblems and with a previously known almost-linear-time algorithm. Additionally, an optimal deterministic algorithm and a linear-time randomized algorithm for sensitivity analysis of minimum spanning trees are presented.

[1]  Bernard Chazelle.  Triangulating a simple polygon in linear time , 1991, Discret. Comput. Geom..

[2]  Michael L. Fredman,et al.  New Bounds on the Complexity of the Shortest Path Problem , 1976, SIAM J. Comput..

[3]  Michael L. Fredman,et al.  Trans-dichotomous algorithms for minimum spanning trees and shortest paths , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[4]  Wayne Goddard,et al.  Optimal randomized algorithms for local sorting and set-maxima , 1990, STOC '90.

[5]  Robert E. Tarjan,et al.  Fast Algorithms for Finding Nearest Common Ancestors , 1984, SIAM J. Comput..

[6]  Robert E. Tarjan,et al.  A linear-time algorithm for a special case of disjoint set union , 1983, J. Comput. Syst. Sci..

[7]  Uzi Vishkin,et al.  On Finding Lowest Common Ancestors: Simplification and Parallelization , 1988, AWOC.

[8]  John H. Reif,et al.  The complexity of elementary algebra and geometry , 1984, STOC '84.

[9]  Lawrence L. Larmore,et al.  An Optimal Algorithm with Unknown Time Complexity for Convex Matrix Searching , 1990, Inf. Process. Lett..

[10]  Ronald L. Graham,et al.  On the History of the Minimum Spanning Tree Problem , 1985, Annals of the History of Computing.

[11]  Dov Harel,et al.  A linear algorithm for finding dominators in flow graphs and related problems , 1985, STOC '85.

[12]  Robert E. Tarjan,et al.  Sensitivity Analysis of Minimum Spanning Trees and Shortest Path Trees , 1982, Inf. Process. Lett..

[13]  Robert E. Tarjan,et al.  Data structures and network algorithms , 1983, CBMS-NSF regional conference series in applied mathematics.

[14]  Robert E. Tarjan,et al.  Efficient algorithms for finding minimum spanning trees in undirected and directed graphs , 1986, Comb..