Clustering Problems on Sliding Windows

We explore clustering problems in the streaming sliding window model in both general metric spaces and Euclidean space. We present the first polylogarithmic space O(1)-approximation to the metric kmedian and metric k-means problems in the sliding window model, answering the main open problem posed by Babcock, Datar, Motwani and O’Callaghan [5], which has remained unanswered for over a decade. Our algorithm uses O(k3 log6 n) space and poly(k, log n) update time. This is an exponential improvement on the space required by the technique due to Babcock, et al. We introduce a data structure that extends smooth histograms as introduced by Braverman and Ostrovsky [8] to operate on a broader class of functions. In particular, we show that using only polylogarithmic space we can maintain a summary of the current window from which we can construct an O(1)-approximate clustering solution. Merge-and-reduce is a generic method in computational geometry for adapting offline algorithms to the in...

• Type: Journal
• Year: 2016
• Where: SODA
• Authors: Vladimir Braverman, Harry Lang, Keith Levin, Morteza Monemizadeh
• Keywords: Algorithms, SODA 2016