Angle-constrained spanners with angle at least π/3

Let S be a set of n points in Rd and let t1 be a real number. A geometric graph G with vertex set S is called a t-spanner for S if for each two points p and q in S, there exists a path Q in G between p and q whose length is at most t times |pq|, the Euclidean distance between p and q. The geometric graph G with vertex set S is called a -angle-constrained t-spanner if G is a t-spanner of S and any two edges {p,q} and {p,r} in G make an angle of at least . In this paper, we show that there is a point set P in the plane such that for every >/3, there is no -angle-constrained t-spanner on P. Moreover, we show that for =/3 and every t /3, there is no -angle-constrained t-spanner on P.Moreover, we show that for =/3 and every t<2/3, there is no -angle-constrained t-spanner on P.

[1]  Giri Narasimhan,et al.  New sparseness results on graph spanners , 1995, Int. J. Comput. Geom. Appl..

[2]  Giri Narasimhan,et al.  Geometric spanner networks , 2007 .

[3]  David Eppstein,et al.  Spanning Trees and Spanners , 2000, Handbook of Computational Geometry.

[4]  Tamás Lukovszki,et al.  New results on geometric spanners and their applications , 1999 .

[5]  Alejandro A. Schäffer,et al.  Graph spanners , 1989, J. Graph Theory.

[6]  Michiel H. M. Smid,et al.  An Optimal Algorithm for Computing Angle-Constrained Spanners , 2010, ISAAC.

[7]  Michiel Smid,et al.  Closest-Point Problems in Computational Geometry , 2000, Handbook of Computational Geometry.

[8]  Paul Chew,et al.  There is a planar graph almost as good as the complete graph , 1986, SCG '86.