a tight unconditional lower bound on distributed randomwalk 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 (?{ 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/( D 3 log n )) 1/4 , performing a random walk of length ?( l ) on an n -node network of diameter D requires ?(?{ l D }+ 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.
lower bound randomwalk time complexity communication complexity distributed algorithms