A Gentle Wakeup Call: Symmetry Breaking with Less Collision Cost
Umesh Biswas, Maxwell Young
TL;DR
This work addresses wakeup on a shared time-slotted channel with no collision detection, focusing on both latency and the cumulative cost of collisions, where each collision costs $\mathcal{C}$. The authors introduce Aim-High (AH), a randomized, window-based algorithm that alternates halving and doubling phases with an initial window $w_0=2^{\mathcal{C}^{\epsilon}}$, achieving sublinear collision-cost when $\mathcal{C}$ is large and polylogarithmic guarantees when it is small. They provide a thorough upper-bound analysis revealing two regimes: $n\sqrt{\mathcal{C}} \le w_0$ yields $\mathbb{E}[\text{latency}] = O(\mathcal{C}^{1/2+2\epsilon})$ and $\mathbb{E}[\text{collisions}] = O(\mathcal{C}^{1/2+\epsilon})$, while $n\sqrt{\mathcal{C}} > w_0$ yields $\mathbb{E}[\text{latency}] = O(\log^{1/(2\epsilon)+5} n)$ and $\mathbb{E}[\text{collisions}] = O(\log^{5+3/(2\epsilon)} n)$. A corresponding lower bound shows a trade-off for fair algorithms between latency and collision cost. The results advance understanding of symmetry-breaking under collision costs and have practical implications for networks where collision delays are substantial.
Abstract
The wakeup problem addresses the fundamental challenge of symmetry breaking. Initially, n devices share a time-slotted multiple access channel, which models wireless communication. A transmission succeeds if exactly one device sends in a slot; if two or more transmit, a collision occurs and none succeed. The goal is to achieve a single successful transmission efficiently. Prior work on wakeup primarily analyzes latency -- the number of slots until the first success. However, in many modern systems, each collision incurs a nontrivial delay, C, which prior analyses neglect. Consequently, although existing algorithms achieve polylogarithmic-in-n latency, they still suffer a delay of Ω(C) due to collisions. Here, we design and analyze a randomized wakeup algorithm, Aim-High. When C is sufficiently large with respect to n, Aim-High has expected latency and expected total cost of collisions that are nearly O(\sqrt{C}); otherwise, both quantities are O(poly{\log n}). Finally, for a well-studied class of algorithms, we establish a trade-off between latency and expected total cost of collisions.