Four-Dimensional Dominance Range Reporting in Linear Space
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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+εn + klog^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(nlog^ε n)) space and answer queries in poly-logarithmic time. For comparison the fastest previously known linear-space or O(nlog^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(loglog n (log n/loglog n)^d-3 + k) time using O(nlog^d-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.
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