On interference among moving sensors and related problems

We show that for any set of $n$ points moving along "simple" trajectories (i.e., each coordinate is described with a polynomial of bounded degree) in $\Re^d$ and any parameter $2 \le k \le n$, one can select a fixed non-empty subset of the points of size $O(k \log k)$, such that the Voronoi diagram of this subset is "balanced" at any given time (i.e., it contains $O(n/k)$ points per cell). We also show that the bound $O(k \log k)$ is near optimal even for the one dimensional case in which points move linearly in time. As applications, we show that one can assign communication radii to the sensors of a network of $n$ moving sensors so that at any given time their interference is $O(\sqrt{n\log n})$. We also show some results in kinetic approximate range counting and kinetic discrepancy. In order to obtain these results, we extend well-known results from $\varepsilon$-net theory to kinetic environments

Scalable k-Means Clustering via Lightweight Coresets

Coresets are compact representations of data sets such that models trained on a coreset are provably competitive with models trained on the full data set. As such, they have been successfully used to scale up clustering models to massive data sets. While existing approaches generally only allow for multiplicative approximation errors, we propose a novel notion of lightweight coresets that allows for both multiplicative and additive errors. We provide a single algorithm to construct lightweight coresets for k-means clustering as well as soft and hard Bregman clustering. The algorithm is substantially faster than existing constructions, embarrassingly parallel, and the resulting coresets are smaller. We further show that the proposed approach naturally generalizes to statistical k-means clustering and that, compared to existing results, it can be used to compute smaller summaries for empirical risk minimization. In extensive experiments, we demonstrate that the proposed algorithm outperforms existing data summarization strategies in practice.Comment: To appear in the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining (KDD

Covering many points with a small-area box

Let $P$ be a set of $n$ points in the plane. We show how to find, for a given integer $k>0$, the smallest-area axis-parallel rectangle that covers $k$ points of $P$ in $O(nk^2 \log n+ n\log^2 n)$ time. We also consider the problem of, given a value $\alpha>0$, covering as many points of $P$ as possible with an axis-parallel rectangle of area at most $\alpha$. For this problem we give a probabilistic $(1-\varepsilon)$-approximation that works in near-linear time: In $O((n/\varepsilon^4)\log^3 n \log (1/\varepsilon))$ time we find an axis-parallel rectangle of area at most $\alpha$ that, with high probability, covers at least $(1-\varepsilon)\mathrm{\kappa^*}$ points, where $\mathrm{\kappa^*}$ is the maximum possible number of points that could be covered

Optimal Approximations Made Easy

The fundamental result of Li, Long, and Srinivasan on approximations of set systems has become a key tool across several communities such as learning theory, algorithms, computational geometry, combinatorics and data analysis. The goal of this paper is to give a modular, self-contained, intuitive proof of this result for finite set systems. The only ingredient we assume is the standard Chernoff's concentration bound. This makes the proof accessible to a wider audience, readers not familiar with techniques from statistical learning theory, and makes it possible to be covered in a single self-contained lecture in a geometry, algorithms or combinatorics course.Comment: 7 page