Affiliation:
1. University of Waterloo, Ontario, Canada
2. University of Waterloo
Abstract
We present three new results on one of the most basic problems in geometric data structures,
2-D orthogonal range counting
. All the results are in the
w
-bit word RAM model.
—It is well known that there are linear-space data structures for 2-D orthogonal range counting with worst-case optimal query time
O
(log
n
/log log
n
). We give an
O
(
n
log log
n
)-space
adaptive
data structure that improves the query time to
O
(log log
n
+ log
k
/log log
n
), where
k
is the output count. When
k
=
O
(1), our bounds match the state of the art for the 2-D orthogonal range emptiness problem [Chan et al., 2011].
—We give an
O
(
n
log log
n
)-space data structure for
approximate
2-D orthogonal range counting that can compute a (1 + δ)-factor approximation to the count in
O
(log log
n
) time for any fixed constant δ > 0. Again, our bounds match the state of the art for the 2-D orthogonal range emptiness problem.
—Last, we consider the 1-D
range selection
problem, where a query in an array involves finding the
k
th least element in a given subarray. This problem is closely related to 2-D 3-sided orthogonal range counting. Recently, Jørgensen and Larsen [2011] presented a linear-space adaptive data structure with query time
O
(log log
n
+ log
k
/log log
n
). We give a new linear-space structure that improves the query time to
O
(1 + log
k
/log log
n
), exactly matching the lower bound proved by Jørgensen and Larsen.
Funder
Natural Sciences and Engineering Research Council of Canada
Danish National Research Foundation
Center for Massive Data Algorithmics
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)
Cited by
5 articles.
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