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Four-Dimensional Dominance Range Reporting in Linear Space

Author Yakov Nekrich

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LIPIcs.SoCG.2020.59.pdf
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ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Data structures design and analysis
Keywords
  • Range searching
  • geometric data structures
  • word RAM

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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+ε} n + k log^ε n) time, where k is the number of reported points, n is the number of points in the data structure, and ε is an arbitrarily small positive constant. Our second data structure uses O(n log^ε 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^ε n)) space and answer queries in poly-logarithmic time. For comparison the fastest previously known linear-space or O(n log^ε n)-space data structure supports queries in O(n^ε + 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+ε} 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.

Cite As Get BibTex

Yakov Nekrich. Four-Dimensional Dominance Range Reporting in Linear Space. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 59:1-59:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.SoCG.2020.59

Author Details

Yakov Nekrich
  • Department of Computer Science, Michigan Technological University, Houghton, MI, USA

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