Self-Stabilizing Algorithms for Posets on a Linear Array

Consider points (x 1 ; y 1) and (x 2 ; y 2) in < 2. We say (x 1 ; y 1) is dominated by (x 2 ; y 2) if x 1 < x 2 and y 1 < y 2. If (x 1 ; y 1) is dominated by (x 2 ; y 2), then we write (x 1 ; y 1) < D (x 2 ; y 2). For any nite subset P < 2 , the partially ordered set (P; D) is called the dominance poset. In this paper we present two distributed algorithms to compute the maximal points of a dominance poset, on a linear array of machines. Our rst algorithm to compute the maximal points of a dominance poset begins with each point initially believing that it is a maximal point. As the algorithm proceeds each point that is not maximal discovers a point that dominates it. Our second algorithm is self-stabilizing in the sense that no particular initial state is assumed. The machines start from arbitrary initial states and converge to a nal state in which all maximal points are identiied. The self-stabilizing nature of our second algorithm makes it tolerant to transient faults.