An optimal algorithm for computing angle-constrained spanners

• Main Authors: Paz Carmi, Michiel Smid
• Format: Article
• Language: English
• Published: Carleton University 2012-11-01
• Series: Journal of Computational Geometry
• Online Access: http://jocg.org/index.php/jocg/article/view/94

Let <em>S</em> be a set of <em>n</em> points in R^<em>d</em> and let <em>t</em>&gt;1 be a real number. A graph <em>G</em>=(<em>S</em>,<em>E</em>) is called a <em>t</em>-spanner for <em>S</em>, if for any two points <em>p</em> and <em>q</em> in <em>S</em>, the shortest-path distance in <em>G</em> between <em>p</em> and<em>q</em> is at most <em>t</em>|<em>pq</em>|, where |<em>pq</em>| denotes the Euclidean distance between <em>p</em> and <em>q</em>. The graph <em>G</em> is called θ-angle-constrained, if any two distinct edges sharing an endpoint make an angle of at least θ. It is shown that, for any θ with 0&lt;θ&lt;π/3, a θ-angle-constrained <em>t</em>-spanner can be computed in <em>O</em>(<em>n</em>log <em>n</em>) time, where <em>t</em> depends only on θ. For values of θ approaching 0, we have<em>t</em>=1 + <em>O</em>(θ).