Space efficient algorithms for directed series–parallel graphs

Abstract

The subclass of directed series–parallel graphs plays an important role in computer science. Whether a given graph is series–parallel is a well studied problem in algorithmic graph theory, for which fast sequential and parallel algorithms have been developed in a sequence of papers. Also methods are known to solve the reachability and the decomposition problem for series–parallel graphs time efficiently. However, no dedicated results have been obtained for the space complexity of these problems when restricted to series–parallel graphs – the topic of this paper. Deterministic algorithms are presented for the recognition, reachability, decomposition and the path counting problem for series–parallel graphs that use only logarithmic space. Since for arbitrary directed graphs reachability and path counting are believed not to be solvable in Logspace, the main contribution of this work are novel deterministic path finding routines that work correctly in series–parallel graphs, and a characterization of series–parallel graphs by forbidden subgraphs that can be tested space-efficiently. The space bounds are best possible, i.e. the decision problem is shown to be L -complete with respect to AC 0 -reductions. They have also implications for the parallel time complexity of these problems when restricted to series–parallel graphs. Finally, we sketch how these results can be generalized to extension of the series–parallel graph family: to graphs with multiple sources or multiple sinks and to the class of minimal vertex series–parallel graphs.