Coresets in dynamic geometric data streams

Abstract

A dynamic geometric data stream consists of a sequence of m insert/delete operations of points from the discrete space 1,…,Δd [26]. We develop streaming (1 + e)-approximation algorithms for k-median, k-means, MaxCut, maximum weighted matching (MaxWM), maximum travelling salesperson (MaxTSP), maximum spanning tree (MaxST), and average distance over dynamic geometric data streams. Our algorithms maintain a small weighted set of points(a coreset) that approximates with probability 2/3 the current point set with respect to the considered problem during the m insert/delete operations of the data stream. They use poly (e-1, log m, log Δ) space and update time per insert/delete operation for constant k and dimension dHaving a coreset one only needs a fast approximation algorithm for the weighted problem to compute a solution quickly. In fact, even an exponential algorithm is sometimes feasible as its running time may still be polynomial in n. For example one can compute in poly(log n, exp(O((1+log (1⁄e)⁄e)d-1))) time a solution to k-median and k-means [21] where n is the size of the current point set and k and d are constants. Finding an implicit solution to MaxCut can be done in poly(log n, exp((1⁄e)O(1))) time. For MaxST and average distance we require poly(log n, e-1) time and for MaxWM we require O(n3) time to do this.