Parametric and Kinetic Minimum Spanning Trees

We consider the parametric minimum spanning tree problem, in which we are given a graph with edge weights that are linear functions of a parameter λ and wish to compute the sequence of minimum spanning trees generated as λ varies. We also consider the kinetic minimum spanning tree problem, in which λ represents time and the graph is subject in addition to changes such as edge insertions, deletions, and modifications of the weight functions as time progresses. We solve both problems in time O(n2/3 log n) per combinatorial change in the tree (or randomized O(n2/3 log n) per change). Our time bounds reduce to O(n1/2 log n) per change (O(n1/2 log n) randomized) for planar graphs or other minor-closed families of graphs, and O(n1/4 log n) per change (O(n1/4 log n) randomized) for planar graphs with weight changes but no insertions or deletions. 1 ∗Center for Geometric Computing, Department of Computer Science, Duke Univ., Durham, NC, 27708-0129; http://www.cs.duke.edu/∼pankaj/; [email protected]. Work supported in part by Army Research Office MURI grant DAAH04-96-1-0013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.-Israeli Binational Science Foundation. Dept. of Information and Computer Science, Univ. of California, Irvine, CA 92697-3425; http://www.ics.uci.edu/∼eppstein/; eppstein@ics. uci.edu. Work supported in part by NSF grant CCR-9258355 and by matching funds from Xerox Corp. Dept. of Computer Science, Stanford Univ., Stanford, CA, 94305; http://graphics.stanford.edu/∼guibas/; [email protected]. Work supported in part by Army Research Office MURI grant DAAH04-96-10007 and NSF grant CCR-9623851 Compaq Systems Research Center, 130 Lytton Ave., Palo Alto, CA, 94301-1044; http://www.research.digital.com/SRC/personal/Monika Henzinger/home.html; [email protected]. 1Copyright 1998 IEEE. Published in the Proceedings of FOCS’98, 8-11 November 1998 in Palo Alto, CA. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works, must be obtained from the IEEE. Contact: Manager, Copyrights and Permissions / IEEE Service Center / 445 Hoes Lane / P.O. Box 1331 / Piscataway, NJ 08855-1331, USA. Telephone: + Intl. 732-562-3966.