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Author, Editor
Author(s):
Alvarez, Victor
Bringmann, Karl
Curticapean, Radu
Ray, Saurabh
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Not MPG Author(s):
Alvarez, Victor
Curticapean, Radu
Editor(s):
Dey, Tamal K.
Whitesides, Sue
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dblp
Not MPII Editor(s):
Dey, Tamal K.
Whitesides, Sue
BibTeX cite key*:
AlvarezBCS12
Title, Booktitle
Title*:
Counting Crossing Free Structures
Booktitle*:
Proceedings of the Twenty-Eight Annual Symposium on Computational Geometry (SCG'12)
Event, URLs
URL of the conference:
http://socg2012.web.unc.edu/
URL for downloading the paper:
http://doi.acm.org/10.1145/2261250.2261259
Event Address*:
Chapel Hill, NC, USA
Language:
English
Event Date*
(no longer used):
Organization:
ACM
Event Start Date:
17 June 2012
Event End Date:
20 June 2012
Publisher
Name*:
ACM
URL:
http://www.acm.org/
Address*:
New York, NY
Type:
Vol, No, Year, pp.
Series:
Volume:
Number:
Month:
Pages:
61-68
Year*:
2012
VG Wort Pages:
ISBN/ISSN:
978-1-4503-1299-8
Sequence Number:
DOI:
10.1145/2261250.2261259
Note, Abstract, ©
(LaTeX) Abstract:
Let P be a set of n points in the plane. A crossing-free structure on P is a straight-edge planar graph with vertex set in P. Examples of crossing-free structures include triangulations of P, and spanning cycles of P, also known as polygonalizations of P, among others. There has been a large amount of research trying to bound the number of such structures. In particular, bounding the number of triangulations spanned by P has received considerable attention. It is currently known that every set of n points has at most O(30^n) and at least Ω(2.43^n) triangulations. However, much less is known about the algorithmic problem of counting crossing-free structures of a given set P. For example, no algorithm for counting triangulations is known that, on all instances, performs faster than enumerating all triangulations. In this paper we develop a general technique for computing the number of crossing-free structures of an input set P. We apply the technique to obtain algorithms for computing the number of triangulations and spanning cycles of P. The running time of our algorithms is upper bounded by n^O(k), where k is the number of onion layers of P. In particular, we show that our algorithm for counting triangulations is not slower than O(3.1414^n). Given that there are several well-studied configurations of points with at least Ω(3.464^n) triangulations, and some even with Ω(8^n) triangulations, our algorithm is the first to asymptotically outperform any enumeration algorithm for such instances. In fact, it is widely believed that any set of n points must have at least Ω(3.464^n) triangulations. If this is true, then our algorithm is strictly sub-linear in the number of triangulations counted. We also show that our techniques are general enough to solve the restricted triangulation counting problem, which we prove to be W[2]-hard in the parameter k. This implies a “no free lunch” result: In order to be fixed-parameter tractable, our general algorithm must rely on additional properties that are specific to the considered class of structures.
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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
{
AlvarezBCS12
,
AUTHOR = {Alvarez, Victor and Bringmann, Karl and Curticapean, Radu and Ray, Saurabh},
EDITOR = {Dey, Tamal K. and Whitesides, Sue},
TITLE = {Counting Crossing Free Structures},
BOOKTITLE = {Proceedings of the Twenty-Eight Annual Symposium on Computational Geometry (SCG'12)},
PUBLISHER = {ACM},
YEAR = {2012},
ORGANIZATION = {ACM},
PAGES = {61--68},
ADDRESS = {Chapel Hill, NC, USA},
ISBN = {978-1-4503-1299-8},
DOI = {10.1145/2261250.2261259},
}
Entry last modified by Anja Becker, 01/25/2013
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Editor(s)
[Library]
Created
12/13/2012 10:25:02 AM
Revision
1.
0.
Editor
Anja Becker
Karl Bringmann
Edit Date
25.01.2013 11:14:41
13.12.2012 10:25:02