Space Exploration via Proximity Search

TL;DR

This paper explores how to perform various computational tasks on a point set in Euclidean space using only nearest-neighbor queries, enabling approximate clustering and convex hull membership testing.

Contribution

It introduces methods for approximate clustering, convex hull decision, and greedy permutation computation solely via proximity search queries.

Findings

Approximate bi-criteria k-center clustering achievable with NN queries

Convex hull membership can be approximately decided using NN queries

Clustering with meaningful proximity queries on cluster centers is feasible

Abstract

We investigate what computational tasks can be performed on a point set in $\Re^d$, if we are only given black-box access to it via nearest-neighbor search. This is a reasonable assumption if the underlying point set is either provided implicitly, or it is stored in a data structure that can answer such queries. In particular, we show the following: (A) One can compute an approximate bi-criteria $k$-center clustering of the point set, and more generally compute a greedy permutation of the point set. (B) One can decide if a query point is (approximately) inside the convex-hull of the point set. We also investigate the problem of clustering the given point set, such that meaningful proximity queries can be carried out on the centers of the clusters, instead of the whole point set.