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.

Skyline Queries with Noisy Comparisons

“…The Ω(n log 2 |C|) lower bound is actually a particular case of stronger bounds from the literature [4,28]. We can also prove it directly from the information-theoretic bound in the noiseless case with d = 2, adapting the argument of Yao: even when items can be dominated by at most one item from C, there are at least |C| n−|C| · |C|!…”

“…The history of worst-case-oriented comparison-based algorithms is interwoven with the history of convex hull algorithms [31,28,4,11]. Kung et al [31] proposed a Divide and Conquer algorithm with worst case complexity O(d 2 n log d−2 n).…”

“…Kirkpatrick and Seidel then adapted the algorithm into an output-sensitive algorithm with complexity O(d 2 n log d−2 k), and with even better complexity when k √ n. These results are optimal when d ∈ {2, 3} in the comparison model [31,45], and even in the algebraic decision tree model [28]. For d = 2 and d = 3 the results were recently strengthened with instance-optimal algorithms [4]. Chan and Lee [11] raised the question of determining the oracle complexity of skyline queries, and investigate the constant factors in the O(n log k) complexity for d = 2 and d = 3.…”

“…Afshani et al [4] prove a refined nH(S) bound in the comparison tree model for d = 2, where H(S) is the entropy-like expression defined as follows. H(S) is the minimal value of k (|S k |/n) log(n/|S k |) for a partition (S k ) k of S such that each subset S k is either a singleton or an axis-aligned box.…”

Exploiting non-uniform query distributions in data structuring problems

“…(i.e., quadrant) queries in time that is optimal (to within a constant factor). Afshani et al [2] subsequently extended this result to four-sided queries.…”