On Constant Factors in Comparison-Based Geometric Algorithms and Data Structures

Abstract

Many standard problems in computational geometry have been solved asymptotically optimally as far as comparison-based algorithms are concerned, but there has been little work focusing on improving the constant factors hidden in big-Oh bounds on the number of comparisons needed. In this thesis, we consider orthogonal-type problems and present a number of results that achieve optimality in the constant factors of the leading terms, including:

Skyline Queries with Noisy Comparisons

“…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. For arbitrary values of d, Sheng and Tao recently showed that skylines can be computed in O(n log d−2 n) computational complexity with a minor adaptation of Kung et al's algorithm [31], thus removing a d 2 factor.…”

“…For d = 3, an upper bound of 2n log 2 n + O(n) comparisons follows from Kung et al's algorithm [31]; a lower constant factor than the naïve sorting approach, but one checks easily that Kung et al's algorithm does not guarantee better constant factors for oracle complexity than naïve sorting beyond d = 3. A bound of n log 2 k + O(n √ log k) was recently established [11], matching the information-theoretic lower bound of n log 2 k comparisons [28].…”

“…The question of computing an asymptotic equivalent for the oracle complexity of skylines beyond d = 2 is actually left open in [11]. For arbitrary large d, a few algorithms nevertheless outperform the naïve sorting approach in terms of computational and (thereby also) oracle complexity when k is small enough.…”

“…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).…”

Skyline Computation with Noisy Comparisons

“…When d ∈ {2, 3}, the problem even admits "instance-optimal" algorithms [4]. [11] investigates the constant factor for the number of comparisons required to compute skyline, when d ∈ {2, 3}. The technique does not seem to generalize to arbitrary dimensions, and the authors ask among open problems whether arbitrary skylines can be computed with fewer than dnlogn comparisons.…”