Brief Announcement: Initialization with Exponentially Fewer Bits

Abstract

One of the most fundamental challenges in radio networks is avoiding collisions during communication. This is especially important in single-hop radio networks, where only a single processor is able to transmit at a time. To overcome this obstacle, the processors, which we assume are indistinguishable to begin with, solve initialization. That is, they each get assigned distinct numbers from 1 to n. Because collisions occur if two or more processors transmit simultaneously, solving initialization requires Ω(n) rounds, and O(n) round protocols have long been known.
These networks are often made up of battery-powered devices, meaning energy is a crucial resource. We therefore consider a model in which nodes are allowed to sleep to save energy at the expense of communicating. Previous work by Nakano and Olariu [Nakano and Olariu, 2000] showed that initialization can be solved in O(log log n) energy using O(log n)-bit messages. We first show that their energy bound is optimal, giving a lower bound that Ω(log log n) sends are needed by any initialization protocol whose round complexity is nearly quasilinear in n. In the more restrictive setting without sender- or receiver-side collision detection (no-CD Model), we show that we can initialize all but a small fraction f of nodes with an energy complexity that depends only on f. As a corollary, we obtain the first constant-energy algorithm for initializing 0.999n nodes.
We then consider the setting in which messages are limited to single bits. In this case, we present a series of improvements that result in an algorithm in which each node sends and receives up to O(log log n) bits - exponentially fewer than all previous work. Ultimately, we show that this can be done in O(n) single-bit rounds, with a worst-case bit complexity of O(log log n) and a node-averaged bit complexity of O(1).