Space-time tradeoffs for approximate spherical range counting

We present space-time tradeoffs for approximate spherical range counting queries. Given a set S of n data points in ℝ ^d along with a positive approximation factor ε, the goal is to preprocess the points so that, given any Euclidean ball B, we can return the number of points of any subset of S that contains all the points within a (1 - ε)-factor contraction of B, but contains no points that lie outside a (1 + ε)-factor expansion of B. In many applications of range searching it is desirable to offer a tradeoff between space and query time. We present here the first such tradeoffs for approximate range counting queries. Given 0 < ε ≤ 1/2 and a parameter γ, where 2 ≤ γ ≤ 1/ε, we show how to construct a data structure of space O(nγ ^dlog(1/ε)) that allows us to answer ε-approximate spherical range counting queries in time O(log(nγ) + 1/(εγ) ^d-1). The data structure can be built in time O(nγ ^dlog(n/ε)log(1/ε)). Here n, ε, and γ are asymptotic quantities, and the dimension d is assumed to be a fixed constant. At one extreme (low space), this yields a data structure of space O(n log(1/ε)) that can answer approximate range queries in time O(log n + (1/ε) ^d-1) which, up to a factor of O(log 1/ε) in space, matches the best known result for approximate spherical range counting queries. At the other extreme (high space), it yields a data structure of space O((n/ε ^d) log(1/ε)) that can answer queries in time O(log n + log 1/ε). This is the fastest known query time for this problem. We also show how to adapt these data structures to the problem of computing an ε-approximation to the kth nearest neighbor, where k is any integer from 1 to n given at query time. The space bounds are identical to the range searching results, and the query time is larger only by a factor of O(1/(εγ)). Our approach is broadly based on methods developed for approximate Voronoi diagrams (AVDs), but it involves a number of significant extensions from the context of nearest neighbor searching to range searching. These include generalizing AVD node-separation properties from leaves to internal nodes of the tree and constructing efficient generator sets through a radial decomposition of space. We have also developed new arguments to analyze the time and space requirements in this more general setting.