Problems complete for deterministic logarithmic space
Journal of Algorithms
Parallel recognition and decomposition of two terminal series parallel graphs
Information and Computation
Parallel algorithms for shared-memory machines
Handbook of theoretical computer science (vol. A)
Computing algebraic formulas using a constant number of registers
SIAM Journal on Computing
An optimal parallel algorithm for formula evaluation
SIAM Journal on Computing
A very hard log-space counting class
Theoretical Computer Science - Special issue on structure in complexity theory
Parallel recognition of series-parallel graphs
Information and Computation
Graph classes: a survey
The Complexity of Planarity Testing
STACS '00 Proceedings of the 17th Annual Symposium on Theoretical Aspects of Computer Science
Parallel Algorithms for Series Parallel Graphs
ESA '96 Proceedings of the Fourth Annual European Symposium on Algorithms
Random walks, universal traversal sequences, and the complexity of maze problems
SFCS '79 Proceedings of the 20th Annual Symposium on Foundations of Computer Science
The directed planar reachability problem
FSTTCS '05 Proceedings of the 25th international conference on Foundations of Software Technology and Theoretical Computer Science
Logspace optimization problems and their approximability properties
FCT'05 Proceedings of the 15th international conference on Fundamentals of Computation Theory
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The subclass of directed series-parallel graphs plays an important role in computer science. To determine whether a graph is series-parallel is a well studied problem in algorithmic graph theory. Fast sequential and parallel algorithms for this problem have been developed in a sequence of papers. For series-parallel graphs methods are also known to solve the reachability and the decomposition problem time efficiently. However, no dedicated results have been obtained for the space complexity of these problems - the topic of this paper. For this special class of graphs, we develop deterministic algorithms for the recognition, reachability, decomposition and the path counting problem that use only logarithmic space. Since for arbitrary directed graphs reachability and path counting are believed not to be solvable in log-space the main contribution of this work are novel deterministic path finding routines that work correctly in series-parallel graphs, and a characterisation of series-parallel graphs by forbidden subgraphs that can be tested space-efficiently. The space bounds are best possible, i.e. the decision problems is shown to be L-complete with respect to AC°-reductions, and they have also implications for the parallel time complexity of series-parallel graphs. Finally, we sketch how these results can be generalised 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.