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Synopsis
Topics In Computational Geometry by John Edward Zolnowsky. This dissertation, submitted to the Department of Computer Science at Stanford University for the degree of Doctor of Philosophy, examines two core problems in computational geometry. The first analyzes the worst-case average performance of nearest-neighbor searches using kd-tree data structures, comparing cyclic, square, and spread variants and establishing both upper and lower bounds for the total work across all points. The second problem presents an algorithm for efficiently intersecting a finite set of half-spaces in three dimensions, achieving O(N log N) time through a divide-and-conquer framework that mirrors Voronoi-type partitioning and an edge-extension procedure. The work discusses historical context, references foundational results, and evaluates trade-offs between preprocessing costs and query times. Overall, the thesis contributes to the theoretical foundations of geometric data structures and three-dimensional linear- and polyhedral-computation methods, with implications for pattern matching, data analysis, and computational geometry practice.
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Topics in Computational Geometry
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