Optimal circle search despite the presence of faulty robots

Abstract

We consider (n,f)-search on a circle, a search problem of a hidden exit on a circle of unit radius for n>1 robots, f of which are faulty. All the robots start at the center of the circle and can move anywhere with maximum speed 1. During the search, robots may communicate wirelessly. All messages transmitted by all robots are tagged with the robots' unique identifiers which cannot be corrupted. The search is considered complete when the exit is found by a non-faulty robot (which must visit its location) and the remaining non-faulty robots know the correct location of the exit.We study two models of faulty robots. First, crash-faulty robots may stop operating as instructed, and thereafter they remain nonfunctional. Second, Byzantine-faulty robots may transmit untrue messages at any time during the search so as to mislead the non-faulty robots, e.g., lie about the location of the exit.When there are only crash fault robots, we provide optimal algorithms for the (n,f)-search problem, with optimal worst-case search completion time 1+(f+1)2πn. Our main technical contribution pertains to optimal algorithms for (n,1)-search with a Byzantine-faulty robot, minimizing the worst-case search completion time, which equals 1+4πn.We also present an algorithm for the mixed case, with one Byzantine and f−1 crash faulty robots with worst-case search completion time 1+2πnf+2sin⁡2πn.