Solving the Canonical Representation and Star System Problems for Proper Circular-Arc Graphs in Log-Space
We present a <i>deterministic</i>, log-space algorithm that solves st-connectivity in undirected graphs. The previous bound on the space complexity of undirected st-connectivity was log<sup>4/3</sup>(ṡ) obtained by Armoni, Ta-Shma, Wigderson and Zhou (JACM 2000). As undirected st-connectivity is complete for the class of problems solvable by symmetric, nondeterministic, log-space computations (the class SL), this algorithm implies that SL = L (where L is the class of problems solvable by deterministic log-space computations). Independent of our work (and using different techniques), Trifonov (STOC 2005) has presented an <i>O</i>(log <i>n</i> log log <i>n</i>)-space, deterministic algorithm for undirected st-connectivity. Our algorithm also implies a way to construct in log-space a <i>fixed</i> sequence of directions that guides a deterministic walk through all of the vertices of any connected graph. Specifically, we give log-space constructible universal-traversal sequences for graphs with restricted labeling and log-space constructible universal-exploration sequences for general graphs.