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, . . . , ∆} d [26]. We develop streaming (1 + )-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( −1 , log m, log ∆) space and update time per insert/delete operation for constant k and dimension d.Having 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/ )/ ) 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/ ) O(1) )) time. For MaxST and average distance we require poly(log n, −1 ) time and for MaxWM we require O(n 3 ) time to do this.

Coresets and Approximate Clustering for Bregman Divergences

“…Coresets in the context of the Euclidean k-median and the Euclidean k-means problem have been known for some time (see [21,3,15,10,13]). A (k, )-coreset for a set P is a small (weighted) set such that for any set C of k cluster centers the (weighted) clustering cost of the coreset is an approximation for the clustering cost of the original set P with relative error at most .…”

Sublinear-time Algorithms

“…Later, standard constant factor approximation algorithms were given that run in time O(n k) (see, e.g., [47,54]). These sublinear-time results have been extended in many different ways, e.g., to efficient data streaming algorithms and very fast algorithms for Euclidean kmedian and also to k-means, see, e.g., [9,13,17,29,38,39,44,45,48]. For another clustering problem, the min-sum k-clustering problem (which is complement to the Max-k-Cut), for the basic case of k = 2, Indyk [42] (see also [41]) gave a (1+ε)-approximation algorithm that runs in time O(2 1/ε O(1) n (log n) O(1) ), which is sublinear in the full input description size.…”

Data Streams: Algorithms and Applications

“…• problems on geometric streams where space is partitioned in one or higher dimensions into geometrically spaced grids [138,96].…”

“…This line of reasoning has led to new geometric results on Turnstile data streams for minimum spanning tree weight estimation, bichromatic matching, facility location and k-median clustering [138,96,95]. Many open problems remain.…”