Space-Efficient and Fast Algorithms for Multidimensional Dominance Reporting and Counting
We present linear-space sub-logarithmic algorithms for handling the 3-dimensional dominance reporting and the 2-dimensional dominance counting problems. Under the RAM model as described in [M. L. Fredman and D. E. Willard. “Surpassing the information theoretic bound with fusion trees”, Journal of Computer and System Sciences, 47:424–436, 1993], our algorithms achieve O(log n/loglog n+f) query time for the 3-dimensional dominance reporting problem, where f is the output size, and O(log n/loglog n) query time for the 2-dimensional dominance counting problem. We extend these results to any constant dimension d ≥ 3, achieving O(n(log n/loglog n)d − 3) space and O((log n/loglog n)d − 2+ f) query time for the reporting case and O(n(log n/loglog n)d − 2) space and O((log n/loglog n)d − 1) query time for the counting case.
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