Geometric Retrieval for Grid Points in the RAM Model

We consider the problem of d-dimensional searching (d ≥ 3) for four query types: range, partial range, exact match and partial match searching. Let N be the number of points, s be the number of keys specified in a partial match and partial range query and t be the number of points retrieved. We present a data structure with worst case time complexities O(t + logd−2 N), O(t + (d − s) + log N), O(d +√logN) and O(t + (d− s) + s√logN) for each of the aforementioned query types respectively. We also present a second, more concrete solution for exact and partial match queries, which achieves the same query time but has different space requirements. The proposed data structures are considered in the RAM model of computation.