Instance-Optimal Geometric Algorithms

Abstract

We prove the existence of an algorithm A for computing 2D or 3D convex hulls that is optimal for every point set in the following sense: for every sequence σ of n points and for every algorithm A ′ in a certain class A , the running time of A on input σ is at most a constant factor times the running time of A ′ on the worst possible permutation of σ for A ′. In fact, we can establish a stronger property: for every sequence σ of points and every algorithm A ′, the running time of A on σ is at most a constant factor times the average running time of A ′ over all permutations of σ. We call algorithms satisfying these properties instance optimal in the order-oblivious and random-order setting. Such instance-optimal algorithms simultaneously subsume output-sensitive algorithms and distribution-dependent average-case algorithms, and all algorithms that do not take advantage of the order of the input or that assume the input are given in a random order. The class A under consideration consists of all algorithms in a decision tree model where the tests involve only multilinear functions with a constant number of arguments. To establish an instance-specific lower bound, we deviate from traditional Ben-Or-style proofs and adopt a new adversary argument. For 2D convex hulls, we prove that a version of the well-known algorithm by Kirkpatrick and Seidel [1986] or Chan, Snoeyink, and Yap [1995] already attains this lower bound. For 3D convex hulls, we propose a new algorithm. We further obtain instance-optimal results for a few other standard problems in computational geometry, such as maxima in 2D and 3D, orthogonal line segment intersection in 2D, finding bichromatic L ∞ -close pairs in 2D, offline orthogonal range searching in 2D, offline dominance reporting in 2D and 3D, offline half-space range reporting in 2D and 3D, and offline point location in 2D. Our framework also reveals a connection to distribution-sensitive data structures and yields new results as a byproduct, for example, on online orthogonal range searching in 2D and online half-space range reporting in 2D and 3D.