STCON in Directed Unique-Path Graphs

We study the problem of space-efficient polynomial-time algorithms for {em directed st-connectivity} (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 emph{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 $tilde O(n^{varepsilon})$ space for any constant $varepsilon in (0,1]$. We also give a polynomial-time, $tilde O(n^varepsilon)$-space algorithm to emph{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 polynomial-time algorithms for STCON in different classes of directed graphs - from slightly sublinear-space algorithms for general graphs to $O(log n)$ space algorithms for trees