We consider the problem of performing a random walk in a distributed network. Given bandwidth constraints, the goal of the problem is to minimize the number of rounds required to obtain a random walk sample. Das Sarma et al. [PODC'10] show that a random walk of length l on a network of diameter D can be performed in Õ(√{l D}+D) time. A major question left open is whether there exists a faster algorithm, especially whether the multiplication of √{l} and √{D} is necessary. In this paper, we show a tight unconditional lower bound on the time complexity of distributed random walk computation. Specifically, we show that for any n, D, and D ≤ l ≤ (n/(D3 log n))1/4, performing a random walk of length Θ(l) on an n-node network of diameter D requires Ω(√{lD}+D) time. This bound is unconditional, i.e., it holds for any (possibly randomized) algorithm. To the best of our knowledge, this is the first lower bound that the diameter plays a role of multiplicative factor. Our bound shows that the algorithm of Das Sarma et al. is time optimal. Our proof technique introduces a new connection between bounded-round communication complexity and distributed algorithm lower bounds with D as a trade-off parameter, strengthening the previous study by Das Sarma et al. [STOC'11]. In particular, we make use of the bounded-round communication complexity of the pointer chasing problem. Our technique can be of independent interest and may be useful in showing non-trivial lower bounds on the complexity of other fundamental distributed computing problems.

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