22nd European Workshop on Computational Geometry Institute of Software Technology 4th FSP-Seminar Industrial Geometry, March 2007 Maximizing Maximal Angles for Plane Straight-Line Graphs O. Aichholzer, T. Hackl, M. Hoffmann, C. Huemer, A. Pór, F. Santos, B. Speckmann, B. Vogtenhuber Graz University of Technology, Austria ETH Zürich, Switzerland Universitat Politècnica de Catalunya, Spain Hungarian Academy of Sciences, Hungary Universidad de Cantabria, Spain TU Eindhoven, Netherlands FSP-Seminar March 2007, Graz

22nd European Workshop on Computational Geometry Institute of Software Technology 4th FSP-Seminar Industrial Geometry, March 2007 Plane Geometric Graphs vertices: –n points in the plane –points in general position edges: –straight lines spanned by vertices (geometric graphs) –no crossings (plane) 1

22nd European Workshop on Computational Geometry Institute of Software Technology 4th FSP-Seminar Industrial Geometry, March 2007 Plane Geometric Graphs 2 perfect matchings

22nd European Workshop on Computational Geometry Institute of Software Technology 4th FSP-Seminar Industrial Geometry, March 2007 Plane Geometric Graphs perfect matchings spanning paths 2

22nd European Workshop on Computational Geometry Institute of Software Technology 4th FSP-Seminar Industrial Geometry, March 2007 Plane Geometric Graphs perfect matchings spanning paths spanning trees 2

22nd European Workshop on Computational Geometry Institute of Software Technology 4th FSP-Seminar Industrial Geometry, March 2007 Plane Geometric Graphs perfect matchings spanning paths spanning trees connected plane graphs 2

22nd European Workshop on Computational Geometry Institute of Software Technology 4th FSP-Seminar Industrial Geometry, March 2007 Plane Geometric Graphs perfect matchings spanning paths spanning trees connected plane graphs spanning cycles 2

22nd European Workshop on Computational Geometry Institute of Software Technology 4th FSP-Seminar Industrial Geometry, March 2007 Plane Geometric Graphs perfect matchings spanning paths spanning trees connected plane graphs spanning cycles triangulations 2

22nd European Workshop on Computational Geometry Institute of Software Technology 4th FSP-Seminar Industrial Geometry, March 2007 Plane Geometric Graphs perfect matchings spanning paths spanning trees connected plane graphs spanning cycles triangulations pseudo-triangulations 2

22nd European Workshop on Computational Geometry Institute of Software Technology 4th FSP-Seminar Industrial Geometry, March 2007 Basic Idea 3 Generalizing the principle of large incident angles of pointed pseudo-triangulations to other classes of plane graphs

22nd European Workshop on Computational Geometry Institute of Software Technology 4th FSP-Seminar Industrial Geometry, March 2007 Pseudo-Triangulations pseudo-triangle –3 convex vertices –concave chains 4

22nd European Workshop on Computational Geometry Institute of Software Technology 4th FSP-Seminar Industrial Geometry, March 2007 Pseudo-Triangulations pseudo-triangle –3 convex vertices –concave chains pseudo-triangulation –convex hull –partitioned into pseudo-triangles pointed: each point has an incident angle of at least 4

22nd European Workshop on Computational Geometry Institute of Software Technology 4th FSP-Seminar Industrial Geometry, March Openness point set S, graph G(S) A point in p S is – open in G(S), if it has an incident angle of at least The graph G is – open, if every point in S is – open in G(S) A class of graphs is – open, if for all point sets S there exists an – open graph G(S) of class 5 p q

22nd European Workshop on Computational Geometry Institute of Software Technology 4th FSP-Seminar Industrial Geometry, March 2007 The Question We know that pointed pseudo-triangulations are – open. Can we generalize this concept to other classes of graphs? Given a class of graphs, Does there exist some angle, such that is – open? If yes, what is the maximal such ? 6

22nd European Workshop on Computational Geometry Institute of Software Technology 4th FSP-Seminar Industrial Geometry, March 2007 Min Max Min Max problem Optimization for class of plane graphs: –true for all sets S, even for the worst –for S: take the best graph G(S) –has to hold for any point p in G(S) –for a point p take the maximum incident angle find maximal for each class: min S max G min p S max a A(p,G) {a} 7

22nd European Workshop on Computational Geometry Institute of Software Technology 4th FSP-Seminar Industrial Geometry, March 2007 Triangulations convex hull points are – open 8

22nd European Workshop on Computational Geometry Institute of Software Technology 4th FSP-Seminar Industrial Geometry, March 2007 Triangulations convex hull points are – open take the convex hull triangulate 8

22nd European Workshop on Computational Geometry Institute of Software Technology 4th FSP-Seminar Industrial Geometry, March 2007 Triangulations triangular convex hull (edges a,b,c) closest point for each edge (a,b,c) hexagon with hull points and closest edge points triangles empty one angle { } 2 /3 choose connect recurse on smaller subproblems a b c a c b 9

22nd European Workshop on Computational Geometry Institute of Software Technology 4th FSP-Seminar Industrial Geometry, March 2007 Triangulations 10 Theorem 1: Triangulations are 2 /3-open. Moreover, this bound is best possible.

22nd European Workshop on Computational Geometry Institute of Software Technology 4th FSP-Seminar Industrial Geometry, March 2007 Spanning Trees not more than 5 /3-open: at least 3 /2-open: at least 5 /3-open: 11

22nd European Workshop on Computational Geometry Institute of Software Technology 4th FSP-Seminar Industrial Geometry, March 2007 Spanning Trees Not more than 5 /3-open: At least 3 /2-open: At least 5 /3-open: –diameter –farthest points –case analysis on angles 11

22nd European Workshop on Computational Geometry Institute of Software Technology 4th FSP-Seminar Industrial Geometry, March 2007 not more than 5 /3-open: at least 3 /2-open: at least 5 /3-open: –diameter –farthest points –case analysis on angles Spanning Trees 11 Theorem 2: (general) Spanning Trees are 5 /3-open, and this bound is best possible.

22nd European Workshop on Computational Geometry Institute of Software Technology 4th FSP-Seminar Industrial Geometry, March 2007 Theorem 3: Spanning Trees with maximum vertex degree of at most 3 are 3 /2-open. Spanning Trees (bounded vertex degree 3) At least 3 /2-open: –start with diameter –assign subsets –recursively take diameters –consider tangents –connect subsets 12

22nd European Workshop on Computational Geometry Institute of Software Technology 4th FSP-Seminar Industrial Geometry, March 2007 Theorem 3: Spanning Trees with maximum vertex degree of at most 3 are 3 /2-open. Moreover, this bound is best possible. Spanning Trees (bounded vertex degree 3) 3 /2 12

22nd European Workshop on Computational Geometry Institute of Software Technology 4th FSP-Seminar Industrial Geometry, March 2007 Theorem 3: Spanning Trees with maximum vertex degree of at most 3 are 3 /2-open. Moreover, this bound is best possible. Corrolary: Connected Graphs with bounded vertex degree of at most n-2 are at most 3 /2-open. Spanning Trees (bounded vertex degree 3) 3 /2 12

22nd European Workshop on Computational Geometry Institute of Software Technology 4th FSP-Seminar Industrial Geometry, March 2007 inner angles (consecutive points): –at most one angle /2 –diameter points: no angle /2 in total (n-2) angles /2 zig-zag spanning paths: –two paths per point –each path counted twice Þin total n zig-zag paths +each inner angle occurs in exactly one zig-zag path at least two zig-zag paths with no angle /2 Theorem 4: Spanning Paths (for convex sets) are 3 /2-open, and this bound is best possible. Spanning Paths (convex point sets) < 13

22nd European Workshop on Computational Geometry Institute of Software Technology 4th FSP-Seminar Industrial Geometry, March 2007 Spanning Paths (general) 1.For every vertex q of the convex hull of S, there exists a 5 /4-open spanning path on S starting at q. 2.For every edge q 1 q 2 of the convex hull of S, there exists a 5 /4-open spanning path on S starting with q 1 q 2. Case analysis over occuring angles Proof by induction over the number of points, (1) and (2) not independent Theorem 4: Spanning Paths are 5 /4-open. 14

22nd European Workshop on Computational Geometry Institute of Software Technology 4th FSP-Seminar Industrial Geometry, March 2007 Conclusion Pointed Pseudo-Triangulations (180°) Perfect Matchings2 (360°) Spanning Cycles (180°) Triangulations 2 /3(120°) Spanning Trees (unbounded)5 /3(300°) Spanning Trees with bounded vertex degree 3 /2(270°) Spanning Paths (convex)3 /2(270°) Spanning Paths (general)5 (225°) 15 5 /4 (225°) – 3 /2 (270°) ???

22nd European Workshop on Computational Geometry Institute of Software Technology 4th FSP-Seminar Industrial Geometry, March 2007 Thanks! Thanks for your attention … Grazie Danke Merci Gracias Efcharisto Dank U wel 16