Greedy Spanners in Euclidean Spaces Admit Sublinear Separators

The greedy spanner in low dimensional Euclidean space is a fundamental geometric construction that has been extensively studied over three decades as it possesses the two most basic properties of a good spanner: constant maximum degree and constant lightness

Distributed link scheduling with constant overhead

This paper proposes a new class of simple, distributed algorithms for scheduling in wireless networks. The algorithms generate new schedules in a distributed manner via simple local changes to existing schedules. The class is parameterized by integers k ≥ 1. We show that algorithm k of our class achieves k/(k +2) of the capacity region, for every k ≥ 1. The algorithms have small and constant worst-case overheads: in particular, algorithm k generates a new schedule using (a) time less than 4k + 2 round-trip times between neighboring nodes in the network, and (b) at most three control transmissions by any given node, for any k. The control signals are explicitly specified, and face the same interference effects as normal data transmissions. Our class of distributed wireless scheduling algorithms are the first ones guaranteed to achieve any fixed fraction of the capacity region while using small and constant overheads that do not scale with network size. The parameter k explicitly captures the tradeoff between control overhead and scheduler throughput performance and provides a tuning knob protocol designers can use to harness this trade-off in practice. 1