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Cover Time and Broadcast Time [article]

Robert Elsässer, Thomas Sauerwald
2009 arXiv   pre-print
We introduce a new technique for bounding the cover time of random walks by relating it to the runtime of randomized broadcast.  ...  Together with our upper bound on R(G), this lower bound strongly confirms the intuition of Chandra et al. for graphs with minimum degree Θ(n), since then the cover time equals the broadcast time multiplied  ...  A wide range of techniques to upper bound the cover time is based on the mixing time of a random walk or the closely related spectral gap.  ... 
arXiv:0902.1735v1 fatcat:imiv5mbpinacdc2me6v5igzike

Many Random Walks Are Faster Than One

NOGA ALON, CHEN AVIN, MICHAL KOUCKÝ, GADY KOZMA, ZVI LOTKER, MARK R. TUTTLE
2011 Combinatorics, probability & computing  
We study the cover time -the expected time required to visit every node in a graph at least once -and we show that for a large collection of interesting graphs, running many random walks in parallel yields  ...  We pose a new and intriguing question motivated by distributed computing regarding random walks on graphs: How long does it take for several independent random walks, starting from the same vertex, to  ...  We prove in Section 6 that on a ring the cover time for k random walks is only a factor of Θ(log k) faster than the cover time for a single random walk.  ... 
doi:10.1017/s0963548311000125 fatcat:4kxjmx6gsngsxmnfts3duyiklu

A Technique for Lower Bounding the Cover Time

David Zuckerman
1992 SIAM Journal on Discrete Mathematics  
A general technique for proving lower bounds on expected covering times of random walks on graphs in terms of expected hitting times between vertices is given.  ...  f(# log V for rapidly mixing walks on vertex transitive graphs, where t denotes the maximum expected hitting time between vertices.  ...  The author thanks Umesh Vazirani for helpful discussions and much assistance in the writing of this paper, and David Aldous for helpful comments and advice.  ... 
doi:10.1137/0405007 fatcat:brnmwgdasbcw5mx4fiqqj6uzei

Many random walks are faster than one

Noga Alon, Chen Avin, Michal Koucky, Gady Kozma, Zvi Lotker, Mark R. Tuttle
2008 Proceedings of the twentieth annual symposium on Parallelism in algorithms and architectures - SPAA '08  
We study the cover time-the expected time required to visit every node in a graph at least once-and we show that for a large collection of interesting graphs, running many random walks in parallel yields  ...  We pose a new and intriguing question motivated by distributed computing regarding random walks on graphs: How long does it take for several independent random walks, starting from the same vertex, to  ...  If the conditions of Theorem 17 do not hold then the cover time and hitting time are on the same order and Theorem 14 gives a trivial (not tight) upper bound.  ... 
doi:10.1145/1378533.1378557 dblp:conf/spaa/AlonAKKLT08 fatcat:2tg5ubnn3zgulbanfml2e4xjzu

Many Random Walks Are Faster Than One [article]

Noga Alon, Chen Avin, Michal Koucky, Gady Kozma, Zvi Lotker, Mark R. Tuttle
2007 arXiv   pre-print
We study the cover time - the expected time required to visit every node in a graph at least once - and we show that for a large collection of interesting graphs, running many random walks in parallel  ...  We pose a new and intriguing question motivated by distributed computing regarding random walks on graphs: How long does it take for several independent random walks, starting from the same vertex, to  ...  If the conditions of Theorem 17 do not hold then the cover time and hitting time are on the same order and Theorem 14 gives a trivial (not tight) upper bound.  ... 
arXiv:0705.0467v2 fatcat:4xajctcl4fh2ffq3mjo5dexrfe

Cover Time and Broadcast Time

Robert Elsässer, Thomas Sauerwald, Marc Herbstritt
2009 Symposium on Theoretical Aspects of Computer Science  
Introduction Upper Bounds on the Cover Time Lower Bounds on the Cover Time Conclusions Randomized Broadcast At time step t = 0 one vertex s has a message (is informed) Robert Elsässer · Cover Time and  ...  Introduction Upper Bounds on the Cover Time Lower Bounds on the Cover Time Conclusions Cover Time vs.  ...  Introduction Upper Bounds on the Cover Time Lower Bounds on the Cover Time Conclusions Random Walk starting from some vertex s , take some incident edge uniformly at random in every step 1, 2, . . . transition  ... 
doi:10.4230/lipics.stacs.2009.1842 dblp:conf/stacs/ElsasserS09 fatcat:kbkqbh5lsffpxhatgiim5lnoe4

Random Walks on Regular and Irregular Graphs

Don Coppersmith, Uriel Feige, James Shearer
1996 SIAM Journal on Discrete Mathematics  
For an undirected graph and an optimal cyclic list of all its vertices, the cyclic cover time is the expected time it takes a simple random walk to travel from vertex to vertex along the list, until it  ...  Other results obtained in the processes of proving the main theorem are a similar characterization of minimum resistance spanning trees of graphs, improved bounds on the cover time of graphs, and a simpli  ...  The above bound is tight for complete graphs. In order to obtain an upper bound on the cyclic cover time, or TSP(G 0 ), we shall bound MST(G 0 ).  ... 
doi:10.1137/s0895480193260595 fatcat:hukbfbeubfhorol5gw25ta6ufm

Multiple Random Walks on Graphs: Mixing Few to Cover Many [article]

Nicolás Rivera, Thomas Sauerwald, John Sylvester
2021 arXiv   pre-print
For example, we prove an unconditional lower bound of Ω((n/k) log n) on the stationary cover time, holding for any n-vertex graph G and any 1 ≤ k =o(nlog n ).  ...  First, we improve and tighten various bounds on the stationary cover time when k random walks start from vertices sampled from the stationary distribution.  ...  We thank Jonathan Hermon for some interesting and useful discussions, and Przemys law Gordinowicz for his feedback on an earlier version of this paper.  ... 
arXiv:2011.07893v2 fatcat:crp4ebe7rbcofedwjpqtxvm5yi

On the cover time of random walks on graphs

Jeff D. Kahn, Nathan Linial, Noam Nisan, Michael E. Saks
1989 Journal of theoretical probability  
That is, we study the expected time needed for a random walk on a finite graph to visit every vertex at least once.  ...  We establish an upper bound of O(n 2) for the expectation of the cover time for regular (or nearly regular) graphs. We prove a lower bound of s log n) for the expected cover time for trees.  ...  INTRODUCTION A random walk on a graph is a very simple discrete time process. A particle starts moving on the vertices of the graph.  ... 
doi:10.1007/bf01048274 fatcat:2y5rd2fjafgsfgv6toj3vmsjkm

The Cover Time of Deterministic Random Walks [chapter]

Tobias Friedrich, Thomas Sauerwald
2010 Lecture Notes in Computer Science  
We present general techniques to derive upper bounds for the vertex and edge cover time and derive matching lower bounds for several important graph classes.  ...  The rotor router model is a popular deterministic analogue of a random walk on a graph. Instead of moving to a random neighbor, the neighbors are served in a fixed order.  ...  Lower Bounds on the Deterministic Cover Time We first prove a general lower bound of Ω(m) on the deterministic cover time for all graphs.  ... 
doi:10.1007/978-3-642-14031-0_16 fatcat:twittojmb5hyzkxmu24dlnz5ci

Page 5298 of Mathematical Reviews Vol. , Issue 94i [page]

1994 Mathematical Reviews  
upper bound for the cover time of symmetric graphs and for the fact that the cover time of the unit cube is @(NlogN), where Hy_, is the (N —1)st harmonic number.  ...  Using the electric approach, we provide some general upper and lower bounds for the expected cover times in terms of the diameter of the graph.  ... 

Tight Bounds for the Cover Times of Random Walks with Heterogeneous Step Lengths

Brieuc Guinard, Amos Korman, Markus Bläser, Christophe Paul
2020 Symposium on Theoretical Aspects of Computer Science  
In this paper, we provide a tight bound for the cover time of such a walk, for every integer k> 1. Specifically, up to lower order polylogarithmic factors, the cover time is n^{1+1/(2k-1)}.  ...  Motivated by biological examples in one-dimensional terrains, a recent paper studied the best cover time on an n-node cycle that can be achieved by a random walk process that uses k step lengths [Boczkowski  ...  28:10 Tight Bounds for the Cover Times of Random Walks informal outline.  ... 
doi:10.4230/lipics.stacs.2020.28 dblp:conf/stacs/GuinardK20 fatcat:saldr6qpxfd35in5dda5xrsvbe

The electrical resistance of a graph captures its commute and cover times

A. K. Chandra, P. Raghavan, W. L. Ruzzo, R. Smolensky
1989 Proceedings of the twenty-first annual ACM symposium on Theory of computing - STOC '89  
As a corollary, the cover time (the expected length of a random walk visiting all vertices) is characterized by the maximum resistance R in the graph to within a factor of log n: mR cover time O(mR log  ...  For many graphs, the bounds on cover time obtained in this manner are better than those obtained from previous techniques such as the eigenvalues of the adjacency matrix.  ...  Research supported in part by the National Science Foundation under grant CCR-8703196; a portion of this work was done while the third author was visiting the IBM T.J. Watson Research Center.  ... 
doi:10.1145/73007.73062 dblp:conf/stoc/ChandraRRST89 fatcat:we44k4x6mnhbhkgyfrqkfccy6a

The Dispersion Time of Random Walks on Finite Graphs

Nicolás Rivera, Thomas Sauerwald, Alexandre Stauffer, John Sylvester
2019 The 31st ACM on Symposium on Parallelism in Algorithms and Architectures - SPAA '19  
Moreover, we derive asymptotic upper and lower bound on the dispersion time for several graph classes, such as cliques, cycles, binary trees, ddimensional grids, hypercubes and expanders.  ...  Most of our bounds are tight up to a multiplicative constant. CCS CONCEPTS • Theory of computation → Random walks and Markov chains.  ...  GENERAL BOUNDS 3.1 Upper bounds The first upper bound we present holds for any graph and only requires knowledge of the maximum hitting time of a random walk between two vertices.  ... 
doi:10.1145/3323165.3323204 dblp:conf/spaa/RiveraSSS19 fatcat:ugqwdpa4onaxjpcmad6s47kozi

The electrical resistance of a graph captures its commute and cover times

Ashok K. Chandra, Prabhakar Raghavan, Walter L. Ruzzo, Roman Smolensky, Prasoon Tiwari
1996 Computational Complexity  
As a corollary, the cover time (the expected length of a random walk visiting all vertices) is characterized by the maximum resistance R in the graph to within a factor of log n: mR cover time O(mR log  ...  For many graphs, the bounds on cover time obtained in this manner are better than those obtained from previous techniques such as the eigenvalues of the adjacency matrix.  ...  Research supported in part by the National Science Foundation under grant CCR-8703196; a portion of this work was done while the third author was visiting the IBM T.J. Watson Research Center.  ... 
doi:10.1007/bf01270385 fatcat:6ukr5q5qrzhsvodlgiecp6qy2u
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