|
|
| Line 1: |
Line 1: |
| A '''geometric spanner''' or a '''''k''-spanner graph''' or a '''''k''-spanner''' was initially introduced as a [[weighted graph]] over a set of points as its vertices which for every pair of vertices has a path between them of weight at most ''k'' times the spatial distance between these points for a fixed ''k''. The parameter ''k'' is called the '''stretch factor''' or '''dilation factor''' of the spanner.<ref>{{citation|first1=Giri|last1=Narasimhan|first2=Michiel|last2=Smid|title=Geometric Spanner Networks|publisher=[[Cambridge University Press]]|year=2007|isbn=0-521-81513-4}}.</ref>
| | I would like to introduce myself to you, I am Jayson Simcox but I don't like when individuals use my full title. My spouse and I live in Mississippi and I love each day living here. Playing badminton is a factor that he is completely addicted to. He is an info officer.<br><br>Here is my page; clairvoyant psychic, [http://chorokdeul.co.kr/index.php?document_srl=324263&mid=customer21 visit web site], |
| | |
| In [[computational geometry]], the concept was first discussed by L.P. Chew in 1986,<ref>{{citation|first=L. Paul|last=Chew|contribution=There is a planar graph almost as good as the complete graph|title=Proc. 2nd Annual Symposium on Computational Geometry|year=1986|pages=169–177|doi=10.1145/10515.10534}}.</ref> although the term "spanner" was not used in the original paper.
| |
| | |
| The notion of [[graph spanner]]s has been known in [[graph theory]]: ''k''-spanners are [[spanning subgraph]]s of graphs with similar dilation property, where distances between graph vertices are defined in [[Glossary of graph theory#Distance|graph-theoretical terms]]. Therefore geometric spanners are graph spanners of [[complete graph]]s [[graph embedding|embedded in the plane]] with edge weights equal to the distances between the embedded vertices in the corresponding metric.
| |
| | |
| Spanners may be used in [[computational geometry]] for solving some [[proximity problems]]. They have also found applications in other areas, such as in [[motion planning]], in [[telecommunication network]]s: network reliability, optimization of [[roaming]] in [[mobile network]]s, etc.
| |
| | |
| ==Greedy Algorithm==
| |
| Iterate over every pair of points in the input, adding the edge between those points whenever there does not exist a path through the graph that is at most ''k'' times the distance between those points. This simple algorithm trivially gives a ''k''-spanner called the greedy spanner.
| |
| | |
| Notice that the greedy spanner contains all nearest neighbors, which takes <math>\Omega (n^2)</math> time to compute, showing that the greedy spanner takes at least as long to compute. <ref>{{cite journal|title=Fast Construction of nets in low-dimensional metrics and their applications.|journal=SIAM Journal on Computing|year=2006|volume=5|issue=35|page=1148-1184|author=S. Har-Peled and M. Mendel.}}</ref>
| |
| | |
| The greedy algorithm was first discovered in 1989 independently by Althöfer<ref>{{cite journal|title=On sparse spanners of weighted graphs.|journal=Discrete & Computational Geometry|year=1993|volume=9|page=81-100|author=I. Althöfer, G. Das, D. P. Dobkin, D. Joseph, and J. Soares.}}</ref> and Bern (unpublished).
| |
| | |
| The fastest known algorithm computes the greedy spanner in <math>O(n^2 \log n)</math> time using <math>O(n^2)</math> space. <ref>{{cite journal|title=Computing the greedy spanner in near-quadratic time.|journal=Algorithmica|year=2010|volume=58|page=711-729|author=P. Bose, P. Carmi, M. Farshi, A. Maheshwari, and M. Smid.}}</ref>
| |
| | |
| ==Research==
| |
| Chew's main result was that for a set of points in the plane there is a [[triangulation]] of this pointset such that for any two points there is a path along the edges of the triangulation with length at most <math>\scriptstyle\sqrt 10</math> the [[Euclidean distance]] between the two points. The result was applied in motion planning for finding reasonable approximations of shortest paths among obstacles.
| |
| | |
| The best upper bound known for the Euclidean [[Delaunay triangulation]] is that it is a <math>\scriptstyle{(4\sqrt{3}/9)\pi} \approx 2.418</math>-spanner for its vertices.<ref>{{citation|first1=J. M.|last1=Keil|first2=C. A.|last2=Gutwin|title=Classes of graphs which approximate the complete Euclidean graph|journal=[[Discrete and Computational Geometry]]|volume=7|issue=1|year=1992|pages=13–28|doi=10.1007/BF02187821}}.</ref>
| |
| The lower bound has been increased from <math>\scriptstyle{{\pi}/2}</math> to just over that, to 1.5846.
| |
| <ref>{{citation|first1=P.|last1=Bose|first2=L.|last2=Devroye|first3=M.|last3=Loeffler|first4=J.|last4=Snoeyink|first5=V.|last5=Verma|
| |
| contribution=The spanning ratio of the Delaunay triangulation is greater than <math>\scriptstyle{\pi/2}</math>|
| |
| title=[[Canadian Conference on Computational Geometry|Proc. 21st Canadian Conf. Computational Geometry]]|year=2009|pages=165–167|location=Vancouver}}.</ref>
| |
| | |
| Finding a ''spanner'' in the Euclidean plane with minimal dilation over ''n'' points with at most ''m'' edges is known to be [[NP-hard]].<ref>{{citation|last1=Klein|first1=Rolf|last2=Kutz|first2=Martin|contribution=Computing Geometric Minimum-Dilation Graphs is NP-Hard|editor1-last=Kaufmann|editor1-first=Michael|editor2-last=Wagner|editor2-first=Dorothea|editor2-link=Dorothea Wagner|title=[[International Symposium on Graph Drawing|Proc. 14th International Symposium in Graph Drawing]], Karlsruhe, Germany, 2006|series=[[Lecture Notes in Computer Science]]|volume=4372|year=2007|publisher=[[Springer Verlag]]|isbn=978-3-540-70903-9|pages=196–207|doi=10.1007/978-3-540-70904-6}}.</ref>
| |
| | |
| ===Sparse spanners===
| |
| Active research is ongoing concerning ''spanners'' with either small vertex degree or a small number of edges.
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| [[Category:Geometric algorithms]]
| |
| [[Category:Geometric graphs]]
| |
I would like to introduce myself to you, I am Jayson Simcox but I don't like when individuals use my full title. My spouse and I live in Mississippi and I love each day living here. Playing badminton is a factor that he is completely addicted to. He is an info officer.
Here is my page; clairvoyant psychic, visit web site,