Storage in Computational Geometry
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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√(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√(log n)) preprocessing time, a constant number of real numbers for storage and O(log n) point location time. Kirkpatrick's algorithm uses O(nlog 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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