论文标题
线性空间中的四维优势范围报告
Four-Dimensional Dominance Range Reporting in Linear Space
论文作者
论文摘要
在本文中，我们研究了具有线性或几乎线性空间使用情况的四维优势范围报告问题和数据结构。我们的结果也可用于回答五个侧面有限的四维查询。 The first data structure presented in this paper uses linear space and answers queries in $O(\log^{1+\varepsilon}n + k\log^{\varepsilon} n)$ time, where $k$ is the number of reported points, $n$ is the number of points in the data structure, and $\varepsilon$ is an arbitrarily small positive constant.我们的第二个数据结构使用$ o（n \ log^{\ varepsilon} n）$ space and Answers查询$ O（\ log n+k）$时间。 这些是此问题的第一个数据结构，该数据结构使用线性（resp。$ o（n \ log^{\ varepsilon} n）$），并在poly-logarithmic时间中回答查询。为了比较，以前最快的线性空间或$ o（n \ log^{\ varepsilon} n）$ - 空间数据结构支持$ O（n^{\ varepsilon} + k）$ TIME中的查询（Bentley和Mauer，1980）。我们的结果可以推广到$ d \ ge 4 $尺寸。例如，我们可以回答$ o（\ log \ log n（\ log n/\ log \ log \ log \ log n）^{d-3} + k）$ o（log \ log n/\ log \ log n）^{d-3} + k）$ time使用$ o（n \ log^{d-4 + \ varepsilon} n）$ space。与先前已知的结果最快相比（Chan，2013年），我们的数据结构将空间使用量减少了$ O（\ log n）$，而无需增加查询时间。
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 $O(\log^{1+\varepsilon}n + k\log^{\varepsilon} n)$ time, where $k$ is the number of reported points, $n$ is the number of points in the data structure, and $\varepsilon$ is an arbitrarily small positive constant. Our second data structure uses $O(n \log^{\varepsilon} n)$ space and answers queries in $O(\log n+k)$ time. These are the first data structures for this problem that use linear (resp. $O(n\log^{\varepsilon} n)$) space and answer queries in poly-logarithmic time. For comparison the fastest previously known linear-space or $O(n\log^{\varepsilon} n)$-space data structure supports queries in $O(n^{\varepsilon} + k)$ time (Bentley and Mauer, 1980). Our results can be generalized to $d\ge 4$ 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 $O(n\log^{d-4+\varepsilon}n)$ 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.
