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 fourdimensional queries that are bounded on five sides. The first data structure presented in this paper uses linear space and answers queries inwhere 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.