Efficient algorithms for finding minimum spanning forests of hierarchically defined graphs

Abstract

In an earlier paper (T. Lengauer, Efficient solution of connectivity problems on hierarchically defined graphs, in Reihe “Theoretische Informatik” No. 24, FB 17, Universität-Gesamthochschule Paderborn, Paderborn, West Germany, 1985. (Excerpts in “Proceedings of the WG '85” ( H. Noltemeier, Ed.) Trauner Verlag, Linz, Austria, 1985)) a hierarchical graph model is defined that allows the exploitation of the hierarchy for the more efficient solution of graph problems on very large graphs. The model is motivated by applications in the design of VLSI circuits. We show how to efficiently find minimum spanning forests in this graph model. We solve decision problems pertaining to minimum spanning forests in almost linear time in the length of the hierarchical description, query problems in time linear in the depth of the hierarchy and we construct minimum spanning forests in space linear in the length of the hierarchical description.