Abstract
The motivating problem for this paper is to find the expected covering time of a random walk on a balanced binary tree withn vertices. Previous upper bounds for general graphs ofO(|V| |E|)(1) andO(|V| |E|/d min)(2) imply an upper bound ofO(n 2). We show an upper bound on general graphs ofO(Δ |E| log |V|), which implies an upper bound ofO(n log2 n). The previous lower bound was Ω(|V| log |V|) for trees.(2) In our main result, we show a lower bound of Ω(|V| (log d max |V|)2) for trees, which yields a lower bound of Ω(n log2 n). We also extend our techniques to show an upper bound for general graphs ofO(max{E πTi} log |V|).
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Zuckerman, D. Covering times of random walks on bounded degree trees and other graphs. J Theor Probab 2, 147–157 (1989). https://doi.org/10.1007/BF01048276
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DOI: https://doi.org/10.1007/BF01048276