2022/07/03 by Bhore, Sujoy, Klute, Fabian, Oostveen, Jelle J.
#Computational Geometry (cs.CG) #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.2207.01108
We 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(α(I)log |I|) bits of memory for a set of intervals I with α(I) being the size of the largest independent set of 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(α(R)log |R|) bits.