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- W4313441744 abstract "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 for worst-case start vertices (posed by Alon, Avin, Kouck'y, Kozma, Lotker, and Tuttle~in 2008) remains an open problem. First, we improve and tighten various bounds on the stationary cover 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 $1 leq k =o(nlog n )$. Secondly, we establish the stationary cover times of multiple walks on several fundamental networks up to constant factors. Thirdly, we present a framework characterising worst-case cover times in terms of stationary cover times and a novel, relaxed notion of mixing time for multiple walks called the partial 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) the worst-case cover times for many networks including expanders, preferential attachment graphs, grids, binary trees and hypercubes." @default.
- W4313441744 created "2023-01-06" @default.
- W4313441744 creator A5017390373 @default.
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- W4313441744 date "2020-11-16" @default.
- W4313441744 modified "2023-10-18" @default.
- W4313441744 title "Multiple Random Walks on Graphs: Mixing Few to Cover Many" @default.
- W4313441744 doi "https://doi.org/10.48550/arxiv.2011.07893" @default.
- W4313441744 hasPublicationYear "2020" @default.
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