Undirected St-connectivity Research Articles

Undirected connectivity in log-space

We present a deterministic , log-space algorithm that solves st-connectivity in undirected graphs. The previous bound on the space complexity of undirected st-connectivity was log 4/3 (⋅) 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 O (log n log log n )-space, deterministic algorithm for undirected st-connectivity. Our algorithm also implies a way to construct in log-space a fixed 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.

An $O(\logn \log\logn)$ Space Algorithm for Undirected st-Connectivity

We present a deterministic $O(\log n \log \log n)$ space algorithm for undirected st-connectivity. It is based on a space-efficient simulation of the deterministic EREW algorithm of Chong and Lam [J. Algorithms, 18 (1995), pp. 378-402], an approach suggested by Prof. Vijaya Ramachandran, and uses the universal exploration sequences for trees constructed by Koucký in [Proceedings of the 16th Annual IEEE Conference on Computational Complexity, 2001, pp. 21-27]. Our result improves the $O(\log^{4/3} n)$ bound of Armoni et al. in [Proceedings of the 20th Annual ACM Symposium on Theory of Computing, 1997, pp. 230-239] and is a big step towards the optimal $O(\log n)$. Independently of our result and using a different set of techniques, the optimal bound was achieved by Reingold in [Proceedings of the 37th Annual ACM Symposium on Theory of Computing, 2005, pp. 376-385].

An interview with Vladimir Trifonov 2005 Danny Lewin best student paper award winner

The 2005 Danny Lewin Prize for the best student paper at STOC was awarded to Vladimir Trifonov, of the University of Texas at Austin, for his paper "An O (log n log log n ) Space Algorithm for Undirected st-Connectivity" [1]. Hal Gabow recently started the tradition of interviewing the Danny Lewin Prize winner and publishing the interview in SIGACT News. (An interview with the 2003 winner Tom Hayes can be found in the September 2003 issue.) I had the pleasure of (electronically) chatting with Vladimir about his work; a transcript of our conversation follows.

Time--Space Tradeoffs For Undirected st-Connectivity on a Graph Automata

Undirected st-connectivity is an important problem in computing. There are algorithms for this problem that use O(n) time and ones that use O(log n) space. The main result of this paper is that, in a very natural structured model, these upper bounds are not simultaneously achievable. Any probabilistic jumping automaton for graphs (JAG) requires either space $\Omega( \log^2 n / \log log n )$ or time $n^{(1 + \Omega ( 1 / \log \log n )) }$ to solve undirected st-connectivity.

Undirecteds-t connectivity in polynomial time and sublinear space

Thes-t connectivity problem for undirected graphs is to decide whether two designated vertices,s andt, are in the same connected component. This paper presents the first known deterministic algorithms solving undirecteds-t connectivity using sublinear space and polynomial time. Our algorithms provide a nearly smooth time-space tradeoff between depth-first search and Savitch's algorithm. Forn vertex,m edge graphs, the simplest of our algorithms uses spaceO(s),n 1/2log2 n≤s≤nlog2 n, and timeO(((m+n)n 2 log2 n)/s). We give a variant of this method that is faster at the higher end of the space spectrum. For example, with space θ(nlogn), its time bound isO((m+n)logn), close to the optimal time for the problem. Another generalization uses less space, but more time: spaceO(λn 1/λlogn), for 2≤λ≤log2 n, and timen O(λ). For constant λ the time remains polynomial.

Symmetric Logspace is Closed Under Complement

We present a logspace, many-one reduction from the undirected st-connectivity problem to its complement. This shows that SL = co - SL.