Approximate Range Searching

The range searching problem is a fundamental problem in computational geometry, with numerous important applications. Most research has focused on solving this problem exactly, but lower bounds show that if linear space is assumed, the problem cannot be solved in polylogarithmic time, except for the case of orthogonal ranges. In this paper we show that if one is willing to allow approximate ranges, then it is possible to do much better. In particular, given a bounded range Q of diameter w and > 0, an approximate range query treats the range as a fuzzy object, meaning that points lying within distance w of the boundary of Q either may or may not be counted. We show that in any fixed dimension d, a set of n points in Rd can be preprocessed in O(n log n) time and O(n) space, such that approximate queries can be answered in O(log n + (1/ )d ) time. The only assumption we make about ranges is that the intersection of a range and a d-dimensional cube can be answered in constant time (de...

• Type: Conference
• Year: 1995
• Where: COMPGEOM
• Authors: Sunil Arya, David M. Mount
• Keywords: Approximate Range, COMPGEOM 1995, Computational Geometry, Discrete Geometry, Range Searching Problem