Deterministic Blind Rendezvous in Cognitive Radio Networks

Blind rendezvous is a fundamental problem in cognitive radio networks. The problem involves a collection of agents (radios) that wish to discover each other (i.e., rendezvous) in the blind setting where there is no shared infrastructure and they initially have no knowledge of each other. Time is divided into discrete slots and spectrum is divided into discrete channels, [n] = 1, 2, ..., n. Each agent may access (or hop on) a single channel in a single time slot and two agents rendezvous when they hop on the same channel in the same time slot. The goal is to design deterministic channel hopping schedules for each agent so as to guarantee rendezvous between any pair of agents with access to overlapping sets of channels. The problem has three complicating considerations: first, the agents are asymmetric, i.e., each agent Ai only has access to a particular subset Si ⊂ [n] of the channels and different agents may have access to different subsets of channels (clearly, two agents can rendezvous only if their channel subsets overlap), second, the agents are synchronous, i.e., they do not possess a common sense of absolute time, so different agents may commence their channel schedules at different times (they do have a common sense of slot duration), lastly, agents are anonymous i.e., they do not possess an identity, and hence the schedule for Ai must depend only on Si. Whether guaranteed blind rendezvous in the asynchronous model was even achievable was an open problem. In a recent breakthrough, two independent sets of authors, Shin et al. (Communications Letters, 2010) and Lin et al. (INFOCOM, 2011), gave the first constructions guaranteeing asynchronous blind rendezvous in O (n2) and O (n3) time, respectively. We present a substantially improved and conceptually simpler construction guaranteeing that any two agents, Ai, Aj, will rendezvous in O (|Si||Sj| log log n) time. Our results are the first that achieve nontrivial dependence on |Si|, the sizes of the sets of available channels. This allows us, for example, to save roughly a quadratic factor over the best previous results in the important case when channel subsets have constant size. We also achieve the best possible bound of O (1) rendezvous time for the symmetric situation, previous works could do no better than O (n). Using techniques from the probabilistic method and Ramsey theory we establish that our construction is nearly optimal: we show both an Ω (|Si||Sj|) lower bound and an Ω(log log n) lower bound when |Si|, |Sj| ≤ n/2.