Degree Four Plane Spanners: Simpler And Better

Let P be a set of n points embedded in the plane, and let C be the complete Euclidean graph whose point-set is P. Each edge in C between two points p; q is realized as the line segment [p q] and is assigned a weight equal to the Euclidean distance vertical bar p q vertical bar. In this paper, we show how to construct in O (n vertical bar g n) time a plane spanner of C of maximum degree at most 4 and of stretch factor at most 20. This improves a long sequence of results on the construction of bounded degree plane spanners of C. Our result matches the smallest known upper bound of 4 by Bonichon et al. on the maximum degree while significantly improving their stretch factor upper bound from 156.82 to 20. The construction of our spanner is based on Delaunay triangulations defined with respect to the equilateral-triangle distance, and uses a different approach than that used by Bonichon et al. Our approach leads to a simple and intuitive construction of a well-structured spanner and reveals useful structural properties of Delaunay triangulations defined with respect to the equilateral-triangle distance.The structure of the constructed spanner implies that when P is in convex position, the maximum degree of the spanner is at most 3. Combining the above degree upper bound with the fact that 3 is a lower bound on the maximum degree of any plane spanner of C when the point-set P is in convex position, the results in this paper give a tight bound of 3 on the maximum degree of plane spanners of C for point-sets in convex position.