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<div>[[File:Yao graph.svg|thumb|right|200px]]<br />
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In [[computational geometry]], the '''Yao graph''', named after [[Andrew Yao]], is a kind of [[geometric spanner]], a weighted [[undirected graph]] connecting a set of [[point (geometry)|geometric points]] with the property that, for every pair of points in the graph, their [[shortest path]] has a length that is within a constant factor of their [[Euclidean distance]].<br />
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The basic idea underlying the two-dimensional Yao graph is to surround each of the given points by equally spaced [[ray (geometry)|rays]], partitioning the plane into sectors with equal angles, and to connect each point to its [[nearest neighbor]] in each of these sectors.<ref>{{cite web|title=Overlay Networks for Wireless Systems|url=http://www.cs.jhu.edu/~scheideler/courses/600.348_F04/lecture_13.pdf}}</ref> Associated with a Yao graph is an integer parameter {{math|''k'' ≥ 6}} which is the number of rays and sectors described above; larger values of {{math|''k''}} produce closer approximations to the Euclidean distance.<ref>{{cite web|title=Simple Topologies|url=http://www.cs.kent.edu/~dfuhry/presentations/simple_topologies.pdf}}</ref> The stretch factor is at most <math>1/(\cos \theta - \sin \theta)</math>, where <math>\theta</math> is the angle of the sectors.<ref name=YaoPaper /> The same idea can be extended to point sets in more than two dimensions, but the number of sectors required grows exponentially with the dimension.<br />
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[[Andrew Yao]] used these graphs to construct high-dimensional [[Euclidean minimum spanning tree]]s.<ref name="YaoPaper">{{citation|first=A. C.|last=Yao|authorlink=Andrew Yao|title=On constructing minimum spanning trees in ''k''-dimensional space and related problems|journal=[[SIAM Journal on Computing]]|volume=11|year=1982|pages=721–736|issue=4|doi=10.1137/0211059}}.</ref><br />
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==See also==<br />
* [[Theta graph]]<br />
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==References==<br />
{{Reflist}}<br />
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{{DEFAULTSORT:Yao Graph}}<br />
[[Category:Computational geometry]]<br />
[[Category:Geometric graph theory]]</div>12.150.171.2