Randomized Time-Space Tradeoffs for Directed Graph Connectivity

We present a spectrum of randomized time-space tradeoffs for solving directed graph connectivity or STCONN in small space. We use a strategy parameterized by a parameter k that uses k pebbles and performs short random walks of length \(n^{\frac{1}{k}}\) using a probabilistic counter. We use this to get a family of algorithms that ranges between log2 n and log n in space and 2\(^{{\rm log^2}n}\) and n n in running time. Our approach allows us to look at Savitch’s algorithm and the random walk algorithm as two extremes of the same basic divide and conquer strategy.

[1]  Piotr Berman,et al.  Lower bounds on graph threading by probabilistic machines , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[2]  Stephen A. Cook,et al.  Space Lower Bounds for Maze Threadability on Restricted Machines , 1980, SIAM J. Comput..

[3]  Noam Nisan,et al.  Undirected connectivity in o(\log1 , 1989 .

[4]  Avi Wigderson,et al.  The Complexity of Graph Connectivity , 1992, MFCS.

[5]  E. Szemerédi,et al.  Undirected connectivity in O(log/sup 1.5/n) space , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[6]  Richard J. Lipton,et al.  Random walks, universal traversal sequences, and the complexity of maze problems , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[7]  Michael E. Saks.  Randomization and derandomization in space-bounded computation , 1996, Proceedings of Computational Complexity (Formerly Structure in Complexity Theory).

[8]  Baruch Schieber,et al.  A sublinear space, polynomial time algorithm for directed s-t connectivity , 1992, [1992] Proceedings of the Seventh Annual Structure in Complexity Theory Conference.

[9]  Avi Wigderson,et al.  An O(log(n)4/3) space algorithm for (s, t) connectivity in undirected graphs , 2000, JACM.

[10]  Uriel Feige.  A Spectrum of Time-Space Trade-Offs for Undirected s-t Connectivity , 1997, J. Comput. Syst. Sci..

[11]  Walter J. Savitch,et al.  Relationships Between Nondeterministic and Deterministic Tape Complexities , 1970, J. Comput. Syst. Sci..

[12]  Janos Simon,et al.  Space-bounded probabilistic turing machine complexity classes are closed under complement (Preliminary Version) , 1981, STOC '81.

[13]  John T. Gill,et al.  Computational complexity of probabilistic Turing machines , 1974, STOC '74.

[14]  Noam Nisan.  Rl <= Sc , 1994, Comput. Complex..

[15]  Noam Nisan,et al.  RL⊆SC , 1992, STOC '92.

[16]  Chung Keung Poon.  Space bounds for graph connectivity problems on node-named JAGs and node-ordered JAGs , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[17]  Chung Keung Poon,et al.  Tight Lower Bounds for st-Connectivity on the NNJAG Model , 1999, SIAM J. Comput..