On Streaming Algorithms for Geometric Independent Set and Clique

Abstract

AbstractWe study the maximum geometric independent set and clique problems in the streaming model. Given a collection of geometric objects arriving in an insertion only stream, the aim is to find a subset such that all objects in the subset are pairwise disjoint or intersect respectively.We show that no constant factor approximation algorithm exists to find a maximum set of independent segments or 2-intervals without using a linear number of bits. Interestingly, our proof only requires a set of segments whose intersection graph is also an interval graph. This reveals an interesting discrepancy between segments and intervals as there does exist a 2-approximation for finding an independent set of intervals that uses only \(O(\alpha (\mathcal I)\log |\mathcal I |)\) bits of memory for a set of intervals \(\mathcal I \) with \(\alpha (\mathcal I)\) being the size of the largest independent set of \(\mathcal I \). On the flipside we show that for the geometric clique problem there is no constant-factor approximation algorithm using less than a linear number of bits even for unit intervals. On the positive side we show that the maximum geometric independent set in a set of axis-aligned unit-height rectangles can be 4-approximated using only \(O(\alpha (\mathcal R)\log |\mathcal R |)\) bits.KeywordsGeometric independent setStreaming algorithmsGeometric intersection graphsCommunication lower bounds