Storage in Computational Geometry

TL;DR

This paper introduces a novel storage method for real numbers that enables efficient retrieval and improves the preprocessing complexity of Kirkpatrick's point location algorithm in computational geometry.

Contribution

It presents a new storage technique that reduces storage requirements and enhances the preprocessing time of a key geometric algorithm.

Findings

Storage of n real numbers in constant space with O(log n) fetch time.

Improved preprocessing time for Kirkpatrick's point location algorithm to O(n√log n).

Maintains O(log n) query time with minimal storage.

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].