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 paper, we consider orthogonal-type problems and present a number of results that achieve optimality in the constant factors of the leading terms, includingan algorithm for the 2D maxima problem that uses $$n\lg h + O(n\sqrt{\lg h})$$nlgh+O(nlgh) comparisons, where $$h$$h denotes the output size; a randomized algorithm for the 3D maxima problem that uses $$n\lg h + O(n\lg ^{2/3} h)$$nlgh+O(nlg2/3h) expected number of comparisons; a randomized algorithm for detecting intersections among a set of orthogonal line segments that uses $$n\lg n + O(n\sqrt{\lg n})$$nlgn+O(nlgn) expected number of comparisons; a data structure for point location among 3D disjoint axis-parallel boxes that can answer queries in $$(3/2)\lg n + O(\lg \lg n)$$(3/2)lgn+O(lglgn) comparisons; a data structure for point location in a 3D box subdivision that can answer queries in $$(4/3)\lg n + O(\sqrt{\lg n})$$(4/3)lgn+O(lgn) comparisons. Some of the results can be adapted to solve nonorthogonal problems, such as 2D convex hulls and general line segment intersection. Our algorithms and data structures use a variety of techniques, including Seidel and Adamy's planar point location method, weighted binary search, and height-optimal BSP trees.