A time- and cost-optimal algorithm for overlap graphs, with applications

In various practical applications it is convenient to associate a certain graph with a family I of intervals. One such graph is the well-known overlap graph of I, whose vertices are the intervals in I, with two intervals connected by and edge if and only if they overlap but neither of them strictly contains the other. The first main contribution of this paper is to propose time- and cost-optimal algorithms for computing the connected components of an overlap graph. The task of computing the connected components of an overlap graph arises in numerous applications including frame control, robot arm manipulation, segmentation of range images, routing, automated surveillance systems, recognizing polygonal configurations, and code generation for parallel machines. We begin by showing that any sequential algorithm that determines the connected components of the overlap graph of a family of n intervals must take Ω(n log n) time in the algebraic tree model. Next, we show that any parallel algorithm for this problem must take Ω (log n) time in the CREW model even if an infinite number of processors and memory cells are available. We then go on to show that both the sequential and the parallel lower bounds are tight by providing matching algorithms running in Θ(n log n) sequential time and Θ(log n) time using n processors in the CREW model, respectively