Distributed algorithms for filling MIS vertices of an arbitrary graph by myopic luminous robots

Abstract

We present the problem of finding a maximal independent set (MIS) (named as MIS Filling problem) of an arbitrary connected graph having n vertices with luminous myopic mobile robots. The robots enter the graph one after another from a particular vertex called the Door and move along the edges of the graph without collision to occupy vertices such that the set of occupied vertices form a maximal independent set.In this paper, we explore two versions of the MIS filling problem, where two separate algorithms are proposed. First version of the problem considers a Single Door where our IND algorithm forms an MIS of size m in O(m2) epochs under an asynchronous scheduler, where an epoch is the smallest time interval in which each participating robot gets activated at least once and performs a Look-Compute-Move cycle. The robots have three hops of visibility range, Δ+8 number of colors, and O(log⁡Δ) bits of persistent storage, where Δ is the maximum degree of the graph. The second version of the problem has Multiple Doors for which we present our MULTIND algorithm that forms an MIS in O(m2) epochs under a semi-synchronous scheduler using robots with five hops of visibility range, Δ+k+7 number of colors, and O(log⁡(Δ+k)) bits of persistent storage, where k is the number of Doors. We also achieve a lower bound of Ω(n) for MIS filling problem with Single Door, where n is the number of vertices of the graph.