On Constant Factors in ComparisonBased Geometric Algorithms and Data Structures
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Abstract
Many standard problems in computational geometry have been solved asymptotically optimally as far as comparisonbased algorithms are concerned, but there has been little work focusing on improving the constant factors hidden in bigOh bounds on the number of comparisons needed. In this paper, we consider orthogonaltype problems and present a number of results that achieve optimality in the constant factors of the leading terms, including 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 heightoptimal BSP trees.

an algorithm for the 2D maxima problem that uses \(n\lg h + O(n\sqrt{\lg h})\) comparisons, where \(h\) denotes the output size;

a randomized algorithm for the 3D maxima problem that uses \(n\lg h + O(n\lg ^{2/3} h)\) expected number of comparisons;

a randomized algorithm for detecting intersections among a set of orthogonal line segments that uses \(n\lg n + O(n\sqrt{\lg n})\) expected number of comparisons;

a data structure for point location among 3D disjoint axisparallel boxes that can answer queries in \((3/2)\lg n + O(\lg \lg n)\) comparisons;

a data structure for point location in a 3D box subdivision that can answer queries in \((4/3)\lg n + O(\sqrt{\lg n})\) comparisons.
Keywords
Comparisonbased algorithms Maxima Convex hull Line segment intersection Point location Binary space partitionReferences
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