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arXiv · 2302.11821

Storage in Computational Geometry

Yijie Han·Sanjeev Saxena
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Abstract

We show that $n$ real numbers can be stored in a constant number of real numbers such that each original real number can be fetched in $O(\log n)$ time. Although our result has implications for many computational geometry problems, we show here, combined with Han's $O(n\sqrt{\log n})$ time real number sorting algorithm [3, arXiv:1801.00776], we can improve the complexity of Kirkpatrick's point location algorithm [8] to $O(n\sqrt{\log n})$ preprocessing time, a constant number of real numbers for storage and $O(\log n)$ point location time. Kirkpatrick's algorithm uses $O(n\log n)$ preprocessing time, $O(n)$ storage and $O(\log n)$ point location time. The complexity results in Kirkpatrick's algorithm was the previous best result. Although Lipton and Tarjan's algorithm [10] predates Kirkpatrick's algorithm and has the same complexity, Kirkpatrick's algorithm is simpler and has a better structure. This paper can be viewed as a companion paper of paper [3, arXiv:1801.00776].

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Yijie Han, Sanjeev Saxena. 2023-02-23. Storage in Computational Geometry. https://arxiv.org/abs/2302.11821

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