A general approach for cache-oblivious range reporting and approximate range counting

Abstract

We present cache-oblivious solutions to two important variants of range searching: range reporting and approximate range counting. Our main contribution is a general approach for constructing cache-oblivious data structures that provide relative ( 1 + ε ) -approximations for a general class of range counting queries. This class includes three-sided range counting in the plane, 3-d dominance counting, and 3-d halfspace range counting. The constructed data structures use linear space and answer queries in the optimal query bound of O ( log B ( N / K ) ) block transfers in the worst case, where K is the number of points in the query range. As a corollary, we also obtain the first approximate 3-d halfspace range counting and 3-d dominance counting data structures with a worst-case query time of O ( log ( N / K ) ) in internal memory. An easy but important consequence of our main result is the existence of O ( N log N ) -space cache-oblivious data structures with an optimal query bound of O ( log B N + K / B ) block transfers for the reporting versions of the above problems. Using standard reductions, these data structures allow us to obtain the first cache-oblivious data structures that use almost linear space and achieve the optimal query bound for circular range reporting and K-nearest neighbour searching in the plane, as well as for orthogonal range reporting in three dimensions.