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

Adrian Kosowski 1, 2 
1 CEPAGE - Algorithmics for computationally intensive applications over wide scale distributed platforms
Université Sciences et Technologies - Bordeaux 1, Inria Bordeaux - Sud-Ouest, École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB), CNRS - Centre National de la Recherche Scientifique : UMR5800
Abstract : In 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.
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Submitted on : Tuesday, July 10, 2012 - 12:15:21 AM
Last modification on : Saturday, June 25, 2022 - 8:29:53 PM
Long-term archiving on: : Thursday, October 11, 2012 - 2:55:07 AM


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  • HAL Id : hal-00685373, version 3
  • ARXIV : 1204.1136



Adrian Kosowski. A $\tilde O(n^2)$ Time-Space Trade-off for Undirected s-t Connectivity. SODA - 24th ACM-SIAM Symposium on Discrete Algorithms, Jan 2013, New Orleans, United States. pp.1873-1883. ⟨hal-00685373v3⟩



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