A O~(n2)\tilde O(n^2) Time-Space Trade-off for Undirected s-t Connectivity

Version 3 makes use of the Metropolis-Hastings walkInternational audienceIn this paper, we make use of the Metropolis-type walks due to Nonaka et al. (2010) to provide a faster solution to the $S$-$T$-connectivity problem in undirected graphs (USTCON). As our main result, we propose a family of randomized algorithms for USTCON which achieves a time-space product of $S\cdot T = \tilde O(n^2)$ in graphs with $n$ nodes and $m$ edges (where the $\tilde O$-notation disregards poly-logarithmic terms). This improves the previously best trade-off of $\tilde O(n m)$, due to Feige (1995). Our algorithm consists in deploying several short Metropolis-type walks, starting from landmark nodes distributed using the scheme of Broder et al. (1994) on a modified input graph. In particular, we obtain an algorithm running in time $\tilde O(n+m)$ which is, in general, more space-efficient than both BFS and DFS. We close the paper by showing how to fine-tune the Metropolis-type walk so as to match the performance parameters (e.g., average hitting time) of the unbiased random walk for any graph, while preserving a worst-case bound of $\tilde O(n^2)$ on cover time

Time-space tradeoffs for undirected graph traversal by graph automata

We investigate time-space tradeoffs for traversing undirected graphs, using a variety of structured models that are all variants of Cook and Rackoff's "Jumping Automata for Graphs". Our strongest tradeoff is a quadratic lower bound on the product of time and space for graph traversal. For example, achieving linear time requires linear space, implying that depth-first search is optimal. Since our bound in fact applies to nondeterministic algorithms for nonconnectivity, it also implies that closure under complementation of nondeterministic space-bounded complexity classes is achieved only at the expense of increased time. To demonstrate that these structured models are realistic, we also investigate their power. In addition to admitting well known algorithms such as depth- first search and random walk, we show that one simple variant of this model is nearly as powerful as a Turing machine. Specifically, for general undirected graph problems, it can simulate a Turing machine with only a constant factor increase in space and a polynomial factor increase in time