2020/03/15 by Nekrich, Yakov
#Computational Geometry (cs.CG) #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.2003.06742
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(log1+εn + klogε 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 logε 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(nlogε n)) space and answer queries in poly-logarithmic time. For comparison the fastest previously known linear-space or O(nlogε n)-space data structure supports queries in O(nε + 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(loglog n (log n/loglog n)d-3 + k) time using O(nlogd-4+εn) 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.