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

Abstract

In varioui practical applications it is convenient to associate a cer I ain graph with a family Z of intervals. One such graph is the well-known overlap graphof Z, whose vertices are the iiitervals in I , with two intervals connected by and edge if imd 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 computiiig the connected components of an overlap graph. The iask of computing the connected components of an overlap graph arises in numerous applications including traffic control, robot arm manipulation, segmentation of range images, routing, automated surveillance systems, recognizing polygonal configurations, and code generation for parallel niachines. We begin by showing that any sequential algo i ithm that determines the connected components of the overlap graph of a family of n intervals must take L?(nlog r j ) time in the algebraic tree model. Next, we show that any parallel algorithm for this problem must take L?(logn'i time in the CREW model even if an infinite numbei of processors and meinoiy cells are available. We then go on to show that both the sequential and the parallel lowei bounds are tight by providing matching algorithms running in @ ( n log n ) sequential time and @(log n ) time using n processors in the CREW model, respectively. At the same time, if the endpoints of the intervals are given in sorted ordLi. our sequential algorithm runs in @ ( n ) time, improving the best previously known result. It is interesting to note t l i i L t even if the endpoints are sorted, R(1og n ) is a time lower hound for solving the problem in the CREW model, regaicllrss of the amount of resources available. As an application, we show t h a t our algorithm affords us a time-and cost-optimal solution to a restricted version of the single row routing problem. The best previously known result for routing a set of nets without interstreet crossings run5 in O(1og n log log n ) time using n processors in the CRCT7I model. By contrast, our algorithm runs in ~