Markovian hitters and the complexity of blind rendezvous

Abstract

We define and construct a novel pseudorandom tool, the Markovian hitter. Given an input sequence of n independent random bits, a Markovian hitter produces a sequence of pseudorandom samples in {0, l}k, in an online fashion, that hits any subset W ⊂ {0, l}k of size ϵ2k with probability ≈ 1-2-(n-k)ϵ. This is comparable to the behavior of truly random samples or classical pseudorandom hitting sets. A Markovian hitter has an additional "Markovian" property of interest: each pseudorandom sample is a function of only the 0(k) most recent bits of the input sequence (of random bits). Such Markovian properties are useful in distributed online settings. In particular, we apply Markovian hitters to obtain a new algorithm for the well-studied blind rendezvous problem for cognitive radios. This is the problem faced by two parties equipped with radios that can access channels in potentially different subsets, Si and S2, of a universe of n channels. Their challenge is to discover each other (by tuning their radios to the same channel at the same time) as quickly as possible. In prior work [3] it was shown that deterministic schedules have a lower bound for rendezvous time of Ω(|S1| · |S2|). We beat this quadratic barrier by utilizing a public source of randomness in conjunction with a Markovian hitter to achieve rendezvous in expected time of Ω (S1| · |S2|). We counterbalance this result by establishing two lower bounds on expected rendezvous time: an (Equation presented) bound for the setting with public randomness, and an Ω(|51| · |S2|) bound in the setting with private randomness but no public randomness, which is a strengthening of the result for deterministic schedules.