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Succinct geometric indexes supporting point location queries

Content Provider	ACM Digital Library
Author	Morin, Pat Bose, Prosenjit He, Meng Maheshwari, Anil Chen, Eric Y.
Copyright Year	2012
Abstract	We propose designing data structures called succinct geometric indexes of negligible space (more precisely, $\textit{o}(\textit{n})$ bits) that support geometric queries in optimal time, by taking advantage of the $\textit{n}$ points in the dataset permuted and stored elsewhere as a sequence. Our first and main result is a succinct geometric index that can answer point location queries, a fundamental problem in computational geometry, on planar triangulations in $\textit{O}(lg$ $\textit{n})$ time. We also design three variants of this index. The first supports point location using lg $\textit{n}$ + 2&sqrt;lg $\textit{n}$ + $\textit{O}(lg$ $^{1/4}$ $\textit{n})$ point-line comparisons. The second supports point location in $\textit{o}(lg$ $\textit{n})$ time when the coordinates are integers bounded by $\textit{U}.$ The last variant can answer point location queries in $\textit{O}(\textit{H}$ + 1) expected time, where $\textit{H}$ is the entropy of the query distribution. These results match the query efficiency of previous point location structures that occupy $\textit{O}(\textit{n})$ words or $\textit{O(n}lg$ $\textit{n})$ bits, while saving drastic amounts of space. We generalize our succinct geometric index to planar subdivisions, and design indexes for other types of queries. Finally, we apply our techniques to design the first implicit data structures that support point location in $O(lg^{2}$ $\textit{n})$ time.
Starting Page	1
Ending Page	26
Page Count	26
File Format	PDF
ISSN	15496325
e-ISSN	15496333
DOI	10.1145/2151171.2151173
Volume Number	8
Issue Number	2
Journal	ACM Transactions on Algorithms (TALG)
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2012-04-25
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Geometric data structures Geometric queries Implicit data structures Membership Planar subdivisions Planar triangulations Point location Succinct data structures Succinct geometric indexes Vertical ray shooting
Content Type	Text
Resource Type	Article
Subject	Mathematics

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