New Algorithms for Subset Query, Partial Match, Orthogonal Range Searching, and Related Problems

Abstract

We consider the subset query problem, defined as follows: given a set $\mathcal{P}$ of N subsets of a universe U, |U| = m, build a data structure, which for any query set Q ⊂ U detects if there is any P ∈ $\mathcal{P}$ such that Q ⊂ P. This is essentially equivalent to the partial match problem and is a fundamental problem in many areas. In this paper we present the first (to our knowledge) algorithms, which achieve non-trivial space and query time bounds for m = W (log N). In particular, we present two algorithms with the following tradeoffs:

$$ --N \cdot 2^{O\left( {m log^2 m\sqrt {c/log N} } \right)} $$

space, and O(N/2^c) time, for any c - Nm^c space and O(mN/c) query time, for any c≤ N We extend these results to the more general problem of orthogonal range searching (both exact and approximate versions), approximate orthogonal range intersection and the exact and approximate versions of the nearest neighbor problem in l∞.