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. The main contribution of our paper 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, 3-d dominance counting, and 3-d halfspace range counting. Our technique allows us to obtain data structures that 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. Using the same technique, 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 near-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.

Cache-Oblivious Range Reporting with Optimal Queries Requires Superlinear Space

“…Combined with counting arguments, this ensured that the point set cannot be represented in linear space while guaranteeing a certain proximity (on disk) of the points reported by each query. The problems we study in this paper allow linear-space or O(N log * N)-space solutions in the I/O model [1,2,8], as well as linear-space cache-oblivious solutions for queries of any fixed output size [3,5,10]. This means the previous techniques are ineffective for our purposes.…”

“…Afshani [1] showed that an optimal query bound can in fact be obtained using linear space, raising the question whether this result can be achieved also in the cache-oblivious model. In [3], we show that the optimal query bound can indeed be achieved by a cache-oblivious data structure, but our data structure uses O(N log N) space.…”

“…Recently, Afshani and Chan [2] described a linear-space data structure with the optimal query bound in internal memory and an O(N log * N)-space data structure that answers queries using O(log B N + K/B) block transfers in the I/O model. In [3], we show how to achieve the optimal query bound in the cache-oblivious model, but using O(N log N) space. Table 1 summarizes these results.…”

Adaptive and Approximate Orthogonal Range Counting

“…Since approximate counting is at least as difficult as deciding emptiness, the ultimate goal is to get bounds matching those of emptiness. For example, for approximate 3-D halfspace range counting, Afshani et al [2,3] (improving earlier results [6,24]) obtained linear-space data structures with O(log n) query time. These algorithms can be adapted to solve the related 3-D 3-sided orthogonal range counting problem (i.e., 3-D dominance counting), which includes 2-D 3-sided range counting as a special case.…”

Cache-oblivious range reporting with optimal queries requires superlinear space

“…All computation has to happen on data in internal memory. The transfer of data between internal and external memory happens in blocks of B consecutive data [3,4,7,11] 2-d dominance N log B N + K/B cache-oblivious, [11] 3-d dominance N log N + K internal memory, [1,20] N log B N + K/B I/O model, [1] N log N log B N + K/B cache-oblivious, [3] 3-d halfspace N log N + K internal memory, [2] N log * N log B N + K/B I/O model, [2] N log N log B N + K/B cache-oblivious, [3] Table 1: A summary of related work on three-sided range reporting, 2-d and 3-d dominance reporting, and 3-d halfspace range reporting with the optimal query bound.…”