Deterministic simulation in LOGSPACE
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
Trading space for time in undirected s-t connectivity
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
The electrical resistance of a graph captures its commute and cover times
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
An optimal randomised logarithmic time connectivity algorithm for the EREW PRAM
Journal of Computer and System Sciences
A spectrum of time-space trade-offs for undirected s-tconnectivity
Journal of Computer and System Sciences - Special issue: papers from the 32nd and 34th annual symposia on foundations of computer science, Oct. 2–4, 1991 and Nov. 3–5, 1993
Fast Connected Components Algorithms for the EREW PRAM
SIAM Journal on Computing
Random walks, universal traversal sequences, and the complexity of maze problems
SFCS '79 Proceedings of the 20th Annual Symposium on Foundations of Computer Science
SFCS '88 Proceedings of the 29th Annual Symposium on Foundations of Computer Science
Multiple Random Walks in Random Regular Graphs
SIAM Journal on Discrete Mathematics
Tight bounds for the cover time of multiple random walks
Theoretical Computer Science
Random walks, interacting particles, dynamic networks: randomness can be helpful
SIROCCO'11 Proceedings of the 18th international conference on Structural information and communication complexity
The multi-agent rotor-router on the ring: a deterministic alternative to parallel random walks
Proceedings of the 2013 ACM symposium on Principles of distributed computing
Coalescing-branching random walks on graphs
Proceedings of the twenty-fifth annual ACM symposium on Parallelism in algorithms and architectures
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A random walk on a graph is a process that explores the graph in a random way: at each step the walk is at a vertex of the graph, and at each step it moves to a uniformly selected neighbor of this vertex. Random walks are extremely useful in computer science and in other fields. A very natural problem that was recently raised by Alon, Avin, Koucky, Kozma, Lotker, and Tuttle (though it was implicit in several previous papers) is to analyze the behavior of k independent walks in comparison with the behavior of a single walk. In particular, Alon et al. showed that in various settings (e.g., for expander graphs), k random walks cover the graph (i.e., visit all its nodes), ***(k )-times faster (in expectation) than a single walk. In other words, in such cases k random walks efficiently "parallelize" a single random walk. Alon et al. also demonstrated that, depending on the specific setting, this "speedup" can vary from logarithmic to exponential in k . In this paper we initiate a more systematic study of multiple random walks. We give lower and upper bounds both on the cover time and on the hitting time (the time it takes to hit one specific node) of multiple random walks. Our study revolves over three alternatives for the starting vertices of the random walks: the worst starting vertices (those who maximize the hitting/cover time), the best starting vertices, and starting vertices selected from the stationary distribution. Among our results, we show that the speedup when starting the walks at the worst vertices cannot be too large - the hitting time cannot improve by more than an O (k ) factor and the cover time cannot improve by more than min {k logn ,k 2} (where n is the number of vertices). These results should be contrasted with the fact that there was no previously known upper-bound on the speedup and that the speedup can even be exponential in k for random starting vertices. Some of these results were independently obtained by Elsässer and Sauerwald (ICALP 2009). We further show that for k that is not too large (as a function of various parameters of the graph), the speedup in cover time is O (k ) even for walks that start from the best vertices (those that minimize the cover time). As a rather surprising corollary of our theorems, we obtain a new bound which relates the cover time C and the mixing time mix of a graph. Specifically, we show that (where m is the number of edges).