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Proceedings Article, Paper
@InProceedings
Beitrag in Tagungsband, Workshop

Author, Editor
Author(s):
Arya, Sunil
Malamatos, Theocharis
Mount, David M.
dblp
dblp
dblp
Not MPG Author(s):
Arya, Sunil
Mount, David M.
Editor(s):
BibTeX cite key*:
AMM2006a
Title, Booktitle
Title*:
On the Importance of Idempotence
Booktitle*:
Proceedings of the 38th Annual ACM Symposium on Theory of Computing, STOC'06
Event, URLs
Conference URL::
Downloading URL:
http://delivery.acm.org/10.1145/1140000/1132598/p564-arya.pdf?key1=1132598&key2=3234908711&coll=&dl=&CFID=15151515&CFTOKEN=6184618
Event Address*:
Seattle, Washington, USA
Language:
English
Event Date*
(no longer used):
Organization:
Event Start Date:
21 May 2006
Event End Date:
23 May 2006
Publisher
Name*:
ACM
URL:
Address*:
New York, NY, USA
Type:
Vol, No, Year, pp.
Series:
Volume:
Number:
Month:
Pages:
564-573
Year*:
2006
VG Wort Pages:
ISBN/ISSN:
1-59593-134-1
Sequence Number:
DOI:
Note, Abstract, ©
(LaTeX) Abstract:
Range searching is among the most fundamental problems in computational geometry. An n-element point set in Rd is given along with an assignment of weights to these points from some commutative semigroup. Subject to a fixed space of possible range shapes, the problem is to preprocess the points so that the total semigroup sum of the points lying within a given query range η can be determined quickly. In the approximate version of the problem we assume that η is bounded, and we are given an approximation parameter ε > 0. We are to determine the semigroup sum of all the points contained within η and may additionally include any of the points lying within distance ε • diam(η) of η's boundar.In this paper we contrast the complexity of range searching based on semigroup properties. A semigroup (S,+) is idempotent if x + x = x for all x ∈ S, and it is integral if for all k ≥ 2, the k-fold sum x + ... + x is not equal to x. For example, (R, min) and (0,1, ∨) are both idempotent, and (N, +) is integral. To date, all upper and lower bounds hold irrespective of the semigroup. We show that semigroup properties do indeed make a difference for both exact and approximate range searching, and in the case of approximate range searching the differences are dramatic.First, we consider exact halfspace range searching. The assumption that the semigroup is integral allows us to improve the best lower bounds in the semigroup arithmetic model. For example, assuming O(n) storage in the plane and ignoring polylog factors, we provide an Ω*(n2/5) lower bound for integral semigroups, improving upon the best lower bound of Ω*(n1/3), thus closing the gap with the O(n1/2) upper bound.We also consider approximate range searching for Euclidean ball ranges. We present lower bounds and nearly matching upper bounds for idempotent semigroups. We also present lower bounds for range searching for integral semigroups, which nearly match existing upper bounds. These bounds show that the advantages afforded by idempotency can result in major improvements. In particular, assuming roughly linear space, the exponent in the ε-dependencies is smaller by a factor of nearly 1/2. All our results are presented in terms of space-time tradeoffs, and our lower and upper bounds match closely throughout the entire spectrum.To our knowledge, our results provide the first proof that semigroup properties affect the computational complexity of range searching in the semigroup arithmetic model. These are the first lower bound results for any approximate geometric retrieval problems. The existence of nearly matching upper bounds, throughout the range of space-time tradeoffs, suggests that we are close to resolving the computational complexity of both idempotent and integral approximate spherical range searching in the semigroup arithmetic model.
URL for the Abstract:
http://portal.acm.org/citation.cfm?id=1132516.1132598&coll=&dl=&type=series&idx=1132516&part=Proceedings&WantType=Proceedings&title=Annual%20ACM%20Symposium%20on%20Theory%20of%20Computing&CFID=15151515&CFTOKEN=6184618
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Access Level:
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Correlation
MPG Unit:
Max-Planck-Institut für Informatik
MPG Subunit:
Algorithms and Complexity Group
Appearance:
MPII WWW Server, MPII FTP Server, MPG publications list, university publications list, working group publication list, Fachbeirat, VG Wort



BibTeX Entry:

@INPROCEEDINGS{AMM2006a,
AUTHOR = {Arya, Sunil and Malamatos, Theocharis and Mount, David M.},
TITLE = {On the Importance of Idempotence},
BOOKTITLE = {Proceedings of the 38th Annual ACM Symposium on Theory of Computing, STOC'06},
PUBLISHER = {ACM},
YEAR = {2006},
PAGES = {564--573},
ADDRESS = {Seattle, Washington, USA},
ISBN = {1-59593-134-1},
}


Entry last modified by Christine Kiesel, 05/02/2007
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Editor(s)
Theocharis Malamatos
Created
11/03/2006 08:09:42 PM
Revisions
7.
6.
5.
4.
3.
Editor(s)
Christine Kiesel
Christine Kiesel
Christine Kiesel
Regina Kraemer
Regina Kraemer
Edit Dates
02.05.2007 10:41:07
02.05.2007 10:39:58
02.05.2007 10:22:57
04/16/2007 09:58:35 AM
01.02.2007 09:42:42