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Algorithms for dynamic geometric problems over data streams

Computing over data streams is a recent phenomenon that is of growing interest in many areas of computer science, including databases, computer networks and theory of algorithms. In this scenario, it is assumed that the algorithm sees the elements of the input one-by-one in arbitrary order, and needs to compute a certain function of the input. However, it does not have enough memory to store the whole input. Therefore, it must maintain a “sketch” of the data. Designing a sketching method for a given problem is a novel and exciting challenge for algorithm design. The initial research in streaming algorithms has focused on computing simple numerical statistics of the input, like median [23], number of distinct elements [11] or frequency moments [1]. More recently, the researchers showed that one can use those algorithms as subroutines to solve more complex problems (e.g., cf. [13]); see the survey [24] for detailed description of the past and recent developments. Still, the scope of algorithmic problems for which stream algorithms exist is not well understood. It is therefore of importance to identify new classes of problems that can be solved in this restricted settings. In this paper we investigate stream algorithms for dynamic geometric problems. Specifically, we present low-storage data structures that maintain approximate solutions to geometric problems, under insertions and deletions of points (this is called a turnstile model in [24]). From the data stream perspective, the stream consists of m operations, each of them is either Add(p) (that adds p to the current