Extremal problems on ray sensor configurations

Abstract

A sensor network is typically modeled as a collection of spatially distributed objects with the same shape. The resilience of a sensor network is the number of sensors that can be removed without disrupting the coverage which is the network’s purpose. We introduce two new extremal problems for networks of one-dimensional sensors (lines, rays, and segments) in the two-dimensional plane, where network coverage means protecting locations from external intruders. (1) How well is it possible to simultaneously protect k locations with n (line/ray/segment)-shaped sensors from up to k attackers? (2) How well is it possible to simultaneously protect k locations with n ray-shaped sensors from a single attacker? We show first that (1) and (2) are questions to be answered separately and provide complete answers for k=2 locations for both questions. We also provide asymptotically tight answers for question (1) when k=3,4 and the locations are in convex position, and provide a general lower bound for question (1) that matches the specific asymptotic results.