STCON in Directed Unique-Path Graphs

We study the problem of space-efficient polynomial-time algorithms for directed stconnectivity (STCON). Given a directed graph G, and a pair of vertices s, t, the STCON problem is to decide if there exists a path from s to t in G. For general graphs, the best polynomial-time algorithm for STCON uses space that is only slightly sublinear. However, for special classes of directed graphs, polynomial-time poly-logarithmic-space algorithms are known for STCON. In this paper, we continue this thread of research and study a class of graphs called unique-path graphs with respect to source s, where there is at most one simple path from s to any vertex in the graph. For these graphs, we give a polynomial-time algorithm that uses Õ(n) space for any constant ε ∈ (0, 1]. We also give a polynomial-time, Õ(n)-space algorithm to recognize unique-path graphs. Unique-path graphs are related to configuration graphs of unambiguous log-space computations, but they can have some directed cycles. Our results may be viewed along the continuum of sublinear-space polynomialtime algorithms for STCON in different classes of directed graphs from slightly sublinear-space algorithms for general graphs toO(log n) space algorithms for trees.