Asynchronous Indulgent Gathering of Opaque Robots

Abstract

We consider the benchmarking problem of gathering in an $(N,f)$-fault system consisting of $N$ robots, of which at most $f$ may fail at any execution. The {\em traditional} version asks $N-f$ non-faulty robots to gather at a point not known a priori. The recently studied {\em indulgent} version asks $N-f$ non-faulty robots to gather at the location of a faulty one. Indulgent gathering is impossible in a $(2,1)$-crash system in the $LUMI$ model under semi-synchrony, even when infinitely many colors and coordinate axis agreement are available. In this paper, we establish four results for indulgent gathering under asynchrony in $LUMI$, considering a {\em mobility fault} model in which a robot crash only impacts its mobility but not the operation of the light. We first show that indulgent gathering is impossible in a $(2,1)$-mobility system using 2-colored lights, and then give an optimal solution w.r.t. the number of colors using 3-colored lights. We then give two deterministic algorithms for an $(N,f)$-mobility system, $f<N$, $N,f$ unknown, that exhibit a nice time-color trade-off: One with time $O(N)$ using 7-colored lights and another 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. Our algorithms for an $(N,f)$-mobility system are the first to be analysed time complexity, can withstand obstructed visibility (opaque robot model) and asynchronous scheduling. Interestingly, our algorithms also solve traditional gathering.

37 pages, 26 figures, and 2 tables