Four-Dimensional Dominance Range Reporting in Linear Space

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.