STCON in Directed Unique-Path Graphs

Abstract

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.