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##
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Cover Time and Broadcast Time
[article]

2009
*
arXiv
*
pre-print

We introduce

arXiv:0902.1735v1
fatcat:imiv5mbpinacdc2me6v5igzike
*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. ...##
###
Many Random Walks Are Faster Than One

2011
*
Combinatorics, probability & computing
*

We study

doi:10.1017/s0963548311000125
fatcat:4kxjmx6gsngsxmnfts3duyiklu
*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*. ...##
###
A Technique for Lower Bounding the Cover Time

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. ...

##
###
Many random walks are faster than one

2008
*
Proceedings of the twentieth annual symposium on Parallelism in algorithms and architectures - SPAA '08
*

We study

doi:10.1145/1378533.1378557
dblp:conf/spaa/AlonAKKLT08
fatcat:2tg5ubnn3zgulbanfml2e4xjzu
*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*. ...##
###
Many Random Walks Are Faster Than One
[article]

2007
*
arXiv
*
pre-print

We study

arXiv:0705.0467v2
fatcat:4xajctcl4fh2ffq3mjo5dexrfe
*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*. ...##
###
Cover Time and Broadcast Time

2009
*
Symposium on Theoretical Aspects of Computer Science
*

Introduction

doi:10.4230/lipics.stacs.2009.1842
dblp:conf/stacs/ElsasserS09
fatcat:kbkqbh5lsffpxhatgiim5lnoe4
*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 ...##
###
Random Walks on Regular and Irregular Graphs

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 ). ...

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

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. ...

##
###
On the cover time of random walks on graphs

1989
*
Journal of theoretical probability
*

That is, we study

doi:10.1007/bf01048274
fatcat:2y5rd2fjafgsfgv6toj3vmsjkm
*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*. ...##
###
The Cover Time of Deterministic Random Walks
[chapter]

2010
*
Lecture Notes in Computer Science
*

We present general techniques to derive

doi:10.1007/978-3-642-14031-0_16
fatcat:twittojmb5hyzkxmu24dlnz5ci
*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*. ...##
###
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

2020
*
Symposium on Theoretical Aspects of Computer Science
*

In this paper, we provide

doi:10.4230/lipics.stacs.2020.28
dblp:conf/stacs/GuinardK20
fatcat:saldr6qpxfd35in5dda5xrsvbe
*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. ...##
###
The electrical resistance of a graph captures its commute and cover times

1989
*
Proceedings of the twenty-first annual ACM symposium on Theory of computing - STOC '89
*

As

doi:10.1145/73007.73062
dblp:conf/stoc/ChandraRRST89
fatcat:we44k4x6mnhbhkgyfrqkfccy6a
*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. ...##
###
The Dispersion Time of Random Walks on Finite Graphs

2019
*
The 31st ACM on Symposium on Parallelism in Algorithms and Architectures - SPAA '19
*

Moreover, we derive asymptotic

doi:10.1145/3323165.3323204
dblp:conf/spaa/RiveraSSS19
fatcat:ugqwdpa4onaxjpcmad6s47kozi
*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. ...##
###
The electrical resistance of a graph captures its commute and cover times

1996
*
Computational Complexity
*

As

doi:10.1007/bf01270385
fatcat:6ukr5q5qrzhsvodlgiecp6qy2u
*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. ...
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