A Tight Unconditional Lower Bound On Distributed Random Walk Computation

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 (O) over tilde(root lD + D) time. A major question left open is whether there exists a faster algorithm, especially whether the multiplication of root l and root 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/(D-3 log n))(1/4), performing a random walk of length circle minus(l) on an n-node network of diameter D requires Omega(root 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.