Approximate range searching |
| |
Authors: | Sunil Arya and David M Mount |
| |
Institution: | a Department of Computer Science, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong b Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MD 20742, USA |
| |
Abstract: | 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
can be preprocessed in O(n+logn) time and O(n) space, such that approximate queries can be answered in O(logn(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 (depending on dimension). For convex ranges, we tighten this to O(logn+(1/)d−1) time. We also present a lower bound for approximate range searching based on partition trees of Ω(logn+(1/)d−1), which implies optimality for convex ranges (assuming fixed dimensions). Finally, we give empirical evidence showing that allowing small relative errors can significantly improve query execution times. |
| |
Keywords: | Approximation algorithms Box-decomposition trees Partition trees Range searching |
本文献已被 ScienceDirect 等数据库收录! |
|