Instance-Optimal Imprecise Convex Hull

Abstract

Imprecise measurements of a point set P = (p1, ..., pn) can be modelled by a family of regions F = (R1, ..., Rn), where each imprecise region Ri contains a unique point pi. A retrieval models an accurate measurement by replacing an imprecise region Ri with its corresponding point pi. We construct the convex hull of an imprecise point set in the plane, where regions in F may be retrieved at unit cost. The goal is to determine the cyclic ordering of the convex hull vertices of P as efficiently as possible. Here, efficiency is interpreted in two ways: (i) minimising the number of retrievals, and (ii) computing each retrieval location quickly. Prior works focused on only one of these two aspects: either minimising retrievals or optimising algorithmic runtime. Our contribution is the first to simultaneously achieve both. Let r(F, P) denote the minimal number of retrievals required by any algorithm to determine the convex hull of P for a given instance (F, P). For a family F of n constant-complexity polygons, our main result is a reconstruction algorithm that performs O(r(F, P)) retrievals in O(r(F, P) log^3 n) time. Compared to previous approaches that achieve optimal retrieval counts, we improve the runtime per retrieval by a exponential factor, from polynomial to polylogarithmic. Compared to near-linear time algorithms, we significantly reduce the number of retrievals used, and broaden the input families to include overlapping regions. We further extend our results to simple k-gons and to pairwise disjoint disks with radii in [1,k], where our runtime scales linearly with k.