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Space-Time Tradeoffs for Emptiness Queries

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Erickson, Jeff
Copyright Year	2000
Abstract	We develop the first nontrivial lower bounds on the complexity of online hyperplane and halfspace emptiness queries. Our lower bounds apply to a general class of geometric range query data structures called partition graphs. Informally, a partition graph is a directed acyclic graph that describes a recursive decomposition of space. We show that any partition graph that supports hyperplane emptiness queries implicitly defines a halfspace range query data structure in the Fredman/Yao semigroup arithmetic model, with the same asymptotic space and time bounds. Thus, results of Brnnimann, Chazelle, and Pach imply that any partition graph of size s that supports hyperplane emptiness queries in time t satisfies the inequality $st^d = \Omega((n/\log n)^{d - (d-1)/(d+1)})$. Using different techniques, we improve previous lower bounds for Hopcroft's problem---Given a set of points and hyperplanes, does any hyperplane contain a point?---in dimensions four and higher. Using this offline result, we show that for online hyperplane emptiness queries, $\Omega(n^d/{\mbox{ polylog }} n)$ space is required to achieve polylogarithmic query time, and $\Omega(n^{(d-1)/d}/{\mbox{ polylog }} n)$ query time is required if only O(n polylog n) space is available. These two lower bounds are optimal up to polylogarithmic factors. For two-dimensional queries, we obtain an optimal continuous tradeoff $st^2=\Omega(n^2)$ between these two extremes. Finally, using a lifting argument, we show that the same lower bounds hold for both offline and online halfspace emptiness queries in ${\mathbb{R}}^{d(d+3)/2}$.
Starting Page	1968
Ending Page	1996
Page Count	29
File Format	PDF
ISSN	00975397
DOI	10.1137/S0097539798337212
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	6
Volume Number	29
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-27
Access Restriction	Subscribed
Subject Keyword	Analysis of algorithms and problem complexity Computer graphics; computational geometry partition graph range searching lower bounds space-time tradeoff
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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