Filling MIS Vertices by Myopic Luminous Robots (2107.04885v2)

Abstract: We present the problem of finding a maximal independent set (MIS) (named as \emph{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 \emph{Door} and disperse along the edges of the graph without collision to occupy vertices such that the set of vertices occupied by the robots is a maximal independent set. We assume the robots have knowledge only about the maximum degree of the graph, denoted by $\Delta$. In this paper, we explore two versions of the problem: the solution to the first version, named as \emph{MIS Filling with Single Door}, works under an asynchronous scheduler using robots with 3 hops of visibility range, $\Delta + 6$ number of colors and $O(\log \Delta)$ bits of persistent storage. The time complexity is measured in terms of epochs and it can be solved in $O(n^2)$ epochs. An epoch is the smallest time interval in which each participating robot gets activated and executes the algorithm at least once. For the second version with $k~ ( > 1)$ \textit{Doors}, named as \emph{MIS Filling with Multiple Doors}, the solution works under a semi-synchronous scheduler using robots with 5 hops of visibility range, $\Delta + k + 6$ number of colors and $O(\log (\Delta + k))$ bits of persistent storage. The problem with multiple Doors can be solved in $O(n^2)$ epochs.