Asynchronous Gathering of Opaque Robots with Mobility Faults

Abstract

We consider the fundamental benchmarking problem of gathering in an $(N,f)$-fault system consisting of $N$ robots, of which at most $f$ might fail at any execution, under asynchrony. Two seminal results established impossibility of a solution in the oblivious robot (OBLOT) model in a $(2,0)$-fault system under semi-synchrony and in a $(3,1)$-Byzantine fault system under asynchrony. Recently, a breakthrough result circumvented the first impossibility result by giving a deterministic algorithm in a $(2,0)$-fault system under asynchrony in the luminous robot (LUMI) model using 2-colored lights. However, a breakthrough result established impossibility of gathering in a $(2,1)$-crash system in the LUMI model under semi-synchrony. In this paper, we consider a {\em mobility fault} model in which a robot crash only impacts it mobility but not the operation of the light. We establish four results under asynchrony in LUMI with the mobility fault model. We show that it is impossible to solve gathering in a $(2,1)$-mobility fault system using 2-colored lights, and then give a solution using 3-colored lights, which is optimal w.r.t. the number of colors. We then consider an $(N,f)$-mobility fault system, $f<N$, both $N,f$ not known, and give two deterministic algorithms that exhibit a nice time-color trade-off: The first with time $O(N)$ using 7-colored lights and the second with time $O(\max\{\ell,f\})$ using 26-colored lights, where $\ell< N$ is the number of distinct convex layers of robot positions in the initial configuration. Interestingly, for $l, f = O(1)$, our result is optimal. Our algorithms for an $(N,f)$-mobility fault system are the first to be analysed time complexity, can withstand obstructed visibility (opaque robot model) and asynchronous scheduling.