Minimum Coresets for Maxima Representation of Multidimensional Data

Abstract

Coresets are succinct summaries of large datasets such that, for a given problem, the solution obtained from a coreset is provably competitive with the solution obtained from the full dataset. As such, coreset-based data summarization techniques have been successfully applied to various problems, e.g., geometric optimization, clustering, and approximate query processing, for scaling them up to massive data. In this paper, we study coresets for the maxima representation of multidimensional data: Given a set P of points in $ \mathbbR ^d $, where d is a small constant, and an error parameter $ \varepsilon \in (0,1) $, a subset $ Q \subseteq P $ is an $ \varepsilon $-coreset for the maxima representation of P iff the maximum of Q is an $ \varepsilon $-approximation of the maximum of P for any vector $ u \in \mathbbR ^d $, where the maximum is taken over the inner products between the set of points (P or Q) and u. We define a novel minimum $\varepsilon$-coreset problem that asks for an $\varepsilon$-coreset of the smallest size for the maxima representation of a point set. For the two-dimensional case, we develop an optimal polynomial-time algorithm for the minimum $ \varepsilon $-coreset problem by transforming it into the shortest-cycle problem in a directed graph. Then, we prove that this problem is NP-hard in three or higher dimensions and present polynomial-time approximation algorithms in an arbitrary fixed dimension. Finally, we provide extensive experimental results on both real and synthetic datasets to demonstrate the superior performance of our proposed algorithms.