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/ log log n + f) query time for the 3-dimensional dominance reporting problem, where f is the output size, and O(log n/ log log 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/ log log n)d−3) space and O((log n/ log log n)d−2+ f) query time for the reporting case and O(n(log n/ log log n)d−2) space and O((log n/ log log n)d−1) query time for the counting case.