Deterministic Sampling and Range Counting in Geometric Data Streams

TL;DR

This paper introduces memory-efficient deterministic algorithms for geometric data streams that guarantee approximation bounds and enable approximate online iceberg queries and robust statistics estimation.

Contribution

The paper develops deterministic algorithms for epsilon-nets and epsilon-approximations in geometric streams with guaranteed bounds, improving over probabilistic methods.

Findings

Deterministic algorithms use polylogarithmic memory for approximation.

Algorithms enable approximate online iceberg queries.

Lower bounds established for non-iceberg geometric queries.

Abstract

We present memory-efficient deterministic algorithms for constructing epsilon-nets and epsilon-approximations of streams of geometric data. Unlike probabilistic approaches, these deterministic samples provide guaranteed bounds on their approximation factors. We show how our deterministic samples can be used to answer approximate online iceberg geometric queries on data streams. We use these techniques to approximate several robust statistics of geometric data streams, including Tukey depth, simplicial depth, regression depth, the Thiel-Sen estimator, and the least median of squares. Our algorithms use only a polylogarithmic amount of memory, provided the desired approximation factors are inverse-polylogarithmic. We also include a lower bound for non-iceberg geometric queries.