Deterministic algorithms for undirected s-t connectivity using polynomial time and sublinear space.

Abstract

The s-t connectivity problem for undirected graphs is to decide whether two designated vertices, s and t, are in the same connected component. This paper presents the rst known deterministic algorithms solving undirected s-t connectivity using sublinear space and polynomial time. There is some evidence that such algorithms are impossible for the analogous problem on directed graphs. Our algorithms provide a nearly smooth time-space tradeo between depthrst search and Savitch's algorithm. For n vertex, m edge graphs, the simplest of our algorithms uses space s, O(n 1=2 logn) s O(n logn), and time O(((m+ n)n 2 log 2 n)=s). We give a variant of this method that is faster at the higher end of the space spectrum. For example, with space (n logn), its time bound is O((m+ n) logn), close to the optimal time for the problem. Another generalization uses less space, but more time: space O((n logn)= ), for 1= logn 1=2, and time n O(1= ) . For constant the time remains polynomial. E-mail addresses: greg@-, ruzzo@cs.washington.edu. Research Supported by NSF Grants CCR-8703196 and CCR-9002891. Portions of this work performed while the authors visited the University of Toronto Computer Science Department, whose hospitality is gratefully acknowledged. An extended abstract of these results appeared in 23rd STOC, 1991 [5].