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. ldquoSurpassing the information theoretic bound with fusion treesrdquo, 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 ge 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.