The Directed Planar Reachability Problem
We investigate the s-t-connectivity problem for directed planar graphs, which is hard for L and is contained in NL but is not known to be complete. We show that this problem is logspace-reducible to its complement, and we show that the problem of searching graphs of genus 1 reduces to the planar case.
We also consider a previously-studied subclass of planar graphs known as grid graphs. We show that the directed planar s-t-connectivity problem reduces to the reachability problem for directed grid graphs.
A special case of the grid-graph reachability problem where no edges are directed from right to left is known as the “layered grid graph reachability problem”. We show that this problem lies in the complexity class UL.
KeywordsPlanar Graph Coarse Grid Outer Face Grid Graph Reachability Problem
Unable to display preview. Download preview PDF.
- [Bar02]Barrington, D.A.M.: Grid graph reachability problems. Talk presented at Dagstuhl Seminar on Complexity of Boolean Functions, Seminar number 02121 (2002)Google Scholar
- [BK78]Blum, M., Kozen, D.: On the power of the compass (or, why mazes are easier to search than graphs). In: IEEE Symposium on Foundations of Computer Science (FOCS), pp. 132–142 (1978)Google Scholar
- [MV00]Mahajan, M., Varadarajan, K.R.: A new NC-algorithm for finding a perfect matching in bipartite planar and small genus graphs. In: ACM Symposium on Theory of Computing (STOC), pp. 351–357 (2000)Google Scholar
- [NTS95]Nisan, N., Ta-Shma, A.: Symmetric Logspace is closed under complement. Chicago Journal of Theoretical Computer Science (1995)Google Scholar
- [Rei05]Reingold, O.: Undirected st-connectivity in log-space. In: Proceedings 37th Symposium on Foundations of Computer Science, pp. 376–385. IEEE Computer Society Press, Los Alamitos (2005)Google Scholar