Paper Info
Title
The space complexity of 2-dimensional approximate range counting
Abstract
We study the problem of 2-dimensional orthogonal range counting with additive error. Given a set P of n points drawn from an n x n grid and an error parameter ε, the goal is to build a data structure, such that for any orthogonal range R, the data structure can return the number of points in P ∩ R with additive error εn. A well-known solution for this problem is the ε-approximation. Informally speaking, an ε-approximation of P is a subset A ⊆ P that allows us to estimate the number of points in P ∩ R by counting the number of points in A ∩ R. It is known that an ε-approximation of size O(1/ε log2.5 1/ε) exists for any P with respect to orthogonal ranges, and the best lower bound is Ω(1/ε log 1/ε). The ε-approximation is a rather restricted data structure, as we are not allowed to store any information other than the coordinates of a subset of points in P. In this paper, we explore what can be achieved without any restriction on the data structure. We first describe a data structure that uses O(1/ε log 1/ε log log 1/ε log n) bits that answers queries with error εn. We then prove a lower bound that any data structure that answers queries with error O(log n) must use Ω(n log n) bits. This lower bound has two consequences: 1) answering queries with error O(log n) is as hard as answering the queries exactly; and 2) our upper bound cannot be improved in general by more than an O(log log 1/ε) factor.
Year
DOI
Venue
2013
10.5555/2627817.2627836
Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms
Keywords
DocType
Volume
algorithms,design,counting problems,approximation,theory
Conference
abs/1207.4382
ISBN
Citations 
PageRank 
978-1-61197-251-1
2
0.39
References 
Authors
13
2
Name
Order
Citations
PageRank
Zhewei Wei121520.07
Ke Yi2165977.79