Multiple random walks on graphs: mixing few to cover many
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| Názov: | Multiple random walks on graphs: mixing few to cover many |
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| Autori: | Rivera, Nicolás, Sauerwald, Thomas, Sylvester, John |
| Prispievatelia: | Bansal, Nikhil, Merelli, Emanuela, Worrell, James, Nicolás Rivera and Thomas Sauerwald and John Sylvester, DSpace at Cambridge pro (8.1) |
| Zdroj: | Combinatorics, Probability and Computing. :1-44 |
| Publication Status: | Preprint |
| Informácie o vydavateľovi: | Cambridge University Press (CUP), 2023. |
| Rok vydania: | 2023 |
| Predmety: | FOS: Computer and information sciences, Discrete Mathematics (cs.DM), G.3, G.2.m, 0102 computer and information sciences, 01 natural sciences, Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.), Random Walks, Random walks on graphs, FOS: Mathematics, Mathematics - Combinatorics, 0101 mathematics, multiple random walks, Markov chains, Probability (math.PR), Cover Time, 05C81, 60J10, 60J20, 68R10, Markov Chains, Markov chains (discrete-time Markov processes on discrete state spaces), cover time, mixing time, Multiple Random walks, Mathematics of computing → Stochastic processes, Theory of computation → Random walks and Markov chains, Combinatorics (math.CO), ddc:004, Mathematics - Probability, Computer Science - Discrete Mathematics |
| Popis: | Random walks on graphs are an essential primitive for many randomised algorithms and stochastic processes. It is natural to ask how much can be gained by running$k$multiple random walks independently and in parallel. Although the cover time of multiple walks has been investigated for many natural networks, the problem of finding a general characterisation of multiple cover times forworst-casestart vertices (posed by Alon, Avin, Koucký, Kozma, Lotker and Tuttle in 2008) remains an open problem. First, we improve and tighten various bounds on thestationarycover time when$k$random walks start from vertices sampled from the stationary distribution. For example, we prove an unconditional lower bound of$\Omega ((n/k) \log n)$on the stationary cover time, holding for any$n$-vertex graph$G$and any$1 \leq k =o(n\log n )$. Secondly, we establish thestationarycover times of multiple walks on several fundamental networks up to constant factors. Thirdly, we present a framework characterisingworst-casecover times in terms ofstationarycover times and a novel, relaxed notion of mixing time for multiple walks called thepartial mixing time. Roughly speaking, the partial mixing time only requires a specific portion of all random walks to be mixed. Using these new concepts, we can establish (or recover) theworst-casecover times for many networks including expanders, preferential attachment graphs, grids, binary trees and hypercubes. |
| Druh dokumentu: | Article Conference object |
| Popis súboru: | application/xml; application/pdf |
| Jazyk: | English |
| ISSN: | 1469-2163 0963-5483 |
| DOI: | 10.1017/s0963548322000372 |
| DOI: | 10.48550/arxiv.2011.07893 |
| DOI: | 10.4230/lipics.icalp.2021.107 |
| Prístupová URL adresa: | http://arxiv.org/abs/2011.07893 https://eprints.gla.ac.uk/250808/1/250808.pdf https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.107 https://www.repository.cam.ac.uk/handle/1810/353686 https://doi.org/10.1017/s0963548322000372 |
| Rights: | CC BY |
| Prístupové číslo: | edsair.doi.dedup.....21e92cf24bdd2aaab497645b41da7239 |
| Databáza: | OpenAIRE |
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Nájsť tento článok vo Web of Science | Abstrakt: | Random walks on graphs are an essential primitive for many randomised algorithms and stochastic processes. It is natural to ask how much can be gained by running$k$multiple random walks independently and in parallel. Although the cover time of multiple walks has been investigated for many natural networks, the problem of finding a general characterisation of multiple cover times forworst-casestart vertices (posed by Alon, Avin, Koucký, Kozma, Lotker and Tuttle in 2008) remains an open problem. First, we improve and tighten various bounds on thestationarycover time when$k$random walks start from vertices sampled from the stationary distribution. For example, we prove an unconditional lower bound of$\Omega ((n/k) \log n)$on the stationary cover time, holding for any$n$-vertex graph$G$and any$1 \leq k =o(n\log n )$. Secondly, we establish thestationarycover times of multiple walks on several fundamental networks up to constant factors. Thirdly, we present a framework characterisingworst-casecover times in terms ofstationarycover times and a novel, relaxed notion of mixing time for multiple walks called thepartial mixing time. Roughly speaking, the partial mixing time only requires a specific portion of all random walks to be mixed. Using these new concepts, we can establish (or recover) theworst-casecover times for many networks including expanders, preferential attachment graphs, grids, binary trees and hypercubes. |
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| ISSN: | 14692163 09635483 |
| DOI: | 10.1017/s0963548322000372 |