Global synchronization of pulse-coupled oscillators on trees

Consider a distributed network on a simple graph $G=(V,E)$ with diameter $d$ and maximum degree $\Delta$, where each node has a phase oscillator revolving on $S^{1}=\mathbb{R}/\mathbb{Z}$ with unit speed. Pulse-coupling is a class of distributed time evolution rule for such networked clock system inspired by biological oscillators, where local interactions between neighboring nodes are event-triggered and do not depend on any kind of global information. In this paper, we propose a novel inhibitory pulse-coupling and prove that arbitrary phase configuration on any tree with maximum degree at most 3 and diameter $d$ synchronizes by time $71d$. We extend this pulse-coupling by letting each oscillator may throttle the input according to an auxiliary state variable. We show that this adaptive pulse-coupling synchronizes arbitrary initial configuration on any finite tree with diameter $d$ by time $98d$. As an application, we obtain a universal randomized distributed clock synchronization algorithm, using $O(\log \Delta)$ memory per node with $O(|V|+(d^{5}+\Delta^{2})\log |V|)$ expected worst case running time.