This article is about geometric graphs defined from nearest neighbors in wedges. For theta graphs in graph theory (three paths with shared endpoints), see Glossary of graph theory.
In computational geometry, the Theta graph, or -graph, is a type of geometric spanner similar to a Yao graph. The basic method of construction involves partitioning the space around each vertex into a set of cones, which themselves partition the remaining vertices of the graph. Like Yao Graphs, a -graph contains at most one edge per cone; where they differ is how that edge is selected. Whereas Yao Graphs will select the nearest vertex according to the metric space of the graph, the -graph defines a fixed ray contained within each cone (conventionally the bisector of the cone) and selects the nearest neighbor with respect to orthogonal projections to that ray. The resulting graph exhibits several good spanner properties.[1]
-graphs were first described by Clarkson[2] in 1987 and independently by Keil[3] in 1988.
^Narasimhan, Giri; Smid, Michiel (2007), Geometric Spanner Networks, Cambridge University Press, ISBN 978-0-521-81513-0.
^K. Clarkson. 1987. Approximation algorithms for shortest path motion planning. In Proceedings of the nineteenth annual ACM symposium on Theory of computing (STOC '87), Alfred V. Aho (Ed.). ACM, New York, NY, USA, 56–65.
^Keil, J. (1988). Approximating the complete Euclidean graph. SWAT 88, 208–213.
computational geometry, the Thetagraph, or Θ {\displaystyle \Theta } -graph, is a type of geometric spanner similar to a Yao graph. The basic method of construction...
Appendix:Glossary of graph theory in Wiktionary, the free dictionary. This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes...
{\displaystyle \mathbf {\Theta } } is a matrix of trainable parameters. In particular, let A {\displaystyle \mathbf {A} } be the graph adjacency matrix: then...
spanner can be computed in near quadratic time. The Thetagraph or Θ {\displaystyle \Theta } -graph belongs to the family of cone-based spanners. The basic...
biased graph is a graph with a list of distinguished circles (edge sets of simple cycles), such that if two circles in the list are contained in a theta graph...
non-pseudoforest graph contains as a subgraph a handcuff, figure 8, or thetagraph; any handcuff or figure 8 graph may be contracted to form a butterfly graph (five-vertex...
In the mathematical discipline of graph theory, the line graph of an undirected graph G is another graph L(G) that represents the adjacencies between edges...
ln 3 ) T {\displaystyle \theta =(\theta _{1},\theta _{2})^{T}=(-\ln 2,\ln 3)^{T}} , so that the probability of every graph y ∈ Y {\displaystyle y\in...
In mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors...
{\displaystyle \ell } . The graphs of two polar functions r = f ( θ ) {\displaystyle r=f(\theta )} and r = g ( θ ) {\displaystyle r=g(\theta )} have possible intersections...
_{\zeta }^{2}} . These graphs yield a power-law distribution for the node degrees. The angular coordinate θ {\displaystyle \theta } of each point/node is...
{\displaystyle {\mathcal {L}}(\theta \mid x)} is the probability of observing data x {\displaystyle x} assuming θ {\displaystyle \theta } is the actual parameter...
In graph theory, a branch of mathematics, list coloring is a type of graph coloring where each vertex can be restricted to a list of allowed colors. It...
{\displaystyle \sin ^{2}(\theta )={\frac {1-\cos(2\theta )}{2}}\qquad \cos ^{2}(\theta )={\frac {1+\cos(2\theta )}{2}}} The graph shows both the sine function...
{\widehat {\theta }}} is defined as e ( x ) = θ ^ ( x ) − θ , {\displaystyle e(x)={\widehat {\theta }}(x)-\theta ,} where θ {\displaystyle \theta } is the...
{\displaystyle p(x,y|\theta )=p(y|\theta )p(x|y,\theta )} by using the chain rule. Each parameter vector θ {\displaystyle \theta } is associated with a...
Global Information