Title: Four-dimensional dominance range reporting in linear space
Abstract: In this paper we study the four-dimensional dominance range reporting problem and present data structures with linear or almost-linear space usage. Our results can be also used to answer four-dimensional queries that are bounded on five sides. The first data structure presented in this paper uses linear space and answers queries in O(log^{1+e} n + k log^e n) time, where k is the number of reported points, n is the number of points in the data structure, and e is an arbitrarily small positive constant. Our second data structure uses O(n log^e n) space and answers queries in O(log n+k) time.
These are the first data structures for this problem that use linear (resp. O(n log^e n)) space and answer queries in poly-logarithmic time. For comparison the fastest previously known linear-space or O(n log^e n)-space data structure supports queries in O(n^e + k) time (Bentley and Mauer, 1980). Our results can be generalized to d ≥ 4 dimensions. For example, we can answer d-dimensional dominance range reporting queries in O(log log n (log n/log log n)^{d-3} + k) time using O(n log^{d-4+e} n) space. Compared to the fastest previously known result (Chan, 2013), our data structure reduces the space usage by O(log n) without increasing the query time.
Publication Year: 2020
Publication Date: 2020-01-01
Language: en
Type: article
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Cited By Count: 2
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