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Maximizing Maximal Angles for Plane Straight-Line Graphs

Maximizing Maximal Angles for Plane Straight-Line Graphs. Spanning Paths in General Position Thomas Hackl* Institute for Software Technology, TU Graz Dagstuhl, 12 th – 16 th March 2007. *supported by the Austrian FWF Joint Research Project “Industrial Geometry”S09205-N12. Co-Authors.

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Maximizing Maximal Angles for Plane Straight-Line Graphs

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  1. Maximizing Maximal Anglesfor Plane Straight-Line Graphs Spanning Paths in General PositionThomas Hackl*Institute for Software Technology, TU GrazDagstuhl, 12th– 16th March 2007 *supported by the Austrian FWF Joint Research Project “Industrial Geometry”S09205-N12.

  2. Co-Authors • Oswin Aichholzer, Graz University of Technology • Michael Hoffmann, ETH Zürich • Clemens Huemer, Universitat Politècnica de Catalunya • Attila Pór, Hungarian Academy of Sciences • Francisco Santos, Universidad de Cantabria • Bettina Speckmann, TU Eindhoven • Birgit Vogtenhuber, Graz University of Technology • Research initiated at the 3rd European Pseudotriangulation Week in Berlin organized by Günter Rote and André Schulz. • Thanks to Sarah Kappes, Hannes Krasser, David Orden, Günter Rote, André Schulz, Ileana Streinu and Louis Theran for valueable discussions.

  3. Motivation • Delaunay triangulation maximizes the minimal angle • Minimum Euclidean spanning tree:each angle is at least p/3 • Pointed pseudotriangulation:every point has an incidentangle bigger than p • Other plane geometric graphs?

  4. Definitions • qpr … counterclockwise angle between pq and pr • we only consider angles between consecutive edges • point sets in general position • -open point / vertex p: at least one angle at p is  • -open graph G(S): each pS is -open • -open class of graphs G: on each set S there exists an -open graph G(S)G r p a q

  5. Min Max Min Max problem • Optimization for class G of plane graphs: • true for all sets S, even for the worst • for S: take the best graph G(S)G • has to hold for any point p in G(S) • for a point p take the maximum incident angle • find maximal  for each class: minS maxGG minpS maxaA(p,G){a}

  6. Type of Graphs • pointed pseudotriangulations: obviously 180°-open • perfect matchings: obviously 360°-open • spanning cycles: convex set  180°-open • triangulations ? • pseudotriangulations • plane graphs (connected) in general • spanning trees ? • cycle-free graphs • spanning paths ?

  7. Triangulation • Every finite point set in general position in the planehas a triangulation that is 2p/3-open and this is thebest possible bound.  ?

  8. Spanning tree • Every finite point set in general position in the planehas a spanning tree that is 5p/3-open and this is thebest possible bound. ? 

  9. Spanning tree(bounded vertex degree) • Every finite point set S in general position in the planehas a 3p/2-open spanning tree T such that every point in S has vertex degree at most k in T, 3  k |S|-2.This is the best possible angle bound. p |S|-1 points

  10. Spanning Path (convex) • Point sets in convex position: • Count the “zig-zag” paths: • two per starting vertex • but counting twice •  n “zig-zag” paths • Count big angles: • only angles between consecutive (in radial order) edges used in “zig-zag” paths • each angle is used in exactly one (unique) “zig-zag” path ! • only one (internal) angle  p/2per vertex • for diameter vertices no (internal) angle  p/2 exists •  (n-2) big angles • at least two 3p/2-open spanning paths exist

  11. Spanning Path (general) Every point set S in general position in the plane has a 5p/4-open spanning path. • For every vertex q of the convex hull of S, there exists a 5p/4-open spanning path on S starting at q.  q,p1,...,pk • For every edge q1q2 of the convex hull of S, there exists a 5p/4-open spanning path on S starting with q1q2.either q1,q2,p1,...,pk or q2,q1,p1,...,pk (q,S) (q1q2,S)

  12. Proof Spanning Path • 5p/4-open spanning path: • the smaller of the two angles around each vertex may be at most 3p/4 • Induction on |S|: • Induction base |S| = 3 is obviously true • Case analysis for the induction step • CH(S) … convex hull of point set S • vertex of CH(S) … extreme point of S

  13. Proof Spanning Path • Case 1 (q,S): K = CH(S\{q}) • Case 1.1 • q lies between the outer normalcones of two consecutive verticesy and z of K • Induction on (yz,S\{q})y,z,p1,...,pk or z,y,p1,...,pk • qyzp/2 < 3p/4 and yzqp/2 < 3p/4 • q,y,z,p1,...,pk or q,z,y,p1,...,pk is 5p/4-open K y q z

  14. Proof Spanning Path • Case 1 (q,S): K = CH(S\{q}) • Case 1.2 • q lies in the outer normalcone of a vertex p of K • y and z of K adjacent to p • angles around p: • qpz + zpy + ypq = 2p • subcases on which angle is the smallest(cases qpz or ypq is smallest are symmetric) K y q p z

  15. Proof Spanning Path • Case 1 (q,S): K = CH(S\{q}) • Case 1.2.1 • zpy 2p/3 < 3p/4 is thesmallest angle around p • w.l.o.g. qpz<ypq (qpz < p) • as q in the normal cone of p:qpzp/2 pzqp/2 < 3p/4 • Path starting with q and z followed by (p,S\{q,z})q,z,p,p1,...,pk is 5p/4-open(zpp1zpy) K y q p z

  16. Proof Spanning Path • Case 1 (q,S): K = CH(S\{q}) • Case 1.2.2 • ypq 2p/3 < 3p/4 is thesmallest angle around p • as q in the normal cone of p:qpz, ypqp/2 qyp <p/2 < 3p/4 • starting with q followed by (py,S\{q})we get a 5p/4-open spanning path, either q,p,y,p1,...,pk or q,y,p,p1,...,pk K y q p z

  17. Proof Spanning Path • Case 2 (q1q2,S): K = CH(S\{q1,q2}) • b and c are neighbors of q1 and q2 on CH(S) • l1 and l2 are lines through q1 and q2, respectively, orthogonal to q1q2 • T is the region bounded by q1q2, l1, l2 and K b K q1 l1 T q2 l2 c

  18. Proof Spanning Path b K • Case 2 (q1q2,S): K = CH(S\{q1,q2}) • Case 2.1 • q2q1b or cq2q1 is < 3p/4 • w.l.o.g. q2q1b < 3p/4 • Induction on (q1,S\{q2})5p/4-open path q1,p1,...,pk on S\{q2} • as q2q1p1q2q1b < 3p/4 • 5p/4-open spanning path q2,q1,p1,...,pk on S q1 l1 T q2 l2 c

  19. Proof Spanning Path b K • Case 2 (q1q2,S): K = CH(S\{q1,q2}) • Case 2.2 • q2q1b and cq2q1 are both  3p/4 • Case 2.2.1 no vertex of K exists in T • Case 2.2.2 a vertex p of K exists in T • Case 2.2.2.1 q1pq2 > p/2 • Case 2.2.2.2 q1pq2p/2 q1 l1 T p q2 l2 c

  20. Proof Spanning Path b K • Case 2 (q1q2,S): K = CH(S\{q1,q2}) • Case 2.2 • q2q1b and cq2q1 are both  3p/4 • Case 2.2.1 no vertex of K exists in T • Edge yz of K in T intersects l1 and l2 • if yq2q1 , q2yz , q2q1z , yzq1 < 3p/4 • then we use Induction on (yz,S\{q1,q2}) and extend to a 5p/4-open spanning path q1,q2,y,z,p1,...,pk or q2,q1,z,y,p1,...,pk on S y q1 l1 T q2 l2 z c

  21. Proof Spanning Path b K • Case 2 (q1q2,S): K = CH(S\{q1,q2}) • Case 2.2 q2q1b , cq2q1 3p/4 • Case 2.2.1 no vertex of K exists in T • yq2q1 and q2q1z < p/2 < 3p/4 • q2yz + cq2y < pas the supporting line of yz intersects q2c • and cq2y p/4 as cq2q1 3p/4 •  q2yz < 3p/4 • and yzq1 < 3p/4 by symmetric arguments • if yq2q1 , q2yz , q2q1z , yzq1 < 3p/4 y q1 l1 T q2 l2 z c    

  22. Proof Spanning Path b K • Case 2 (q1q2,S): K = CH(S\{q1,q2}) • Case 2.2.2 a vertex p of K exists in T • q2q1p and pq2q1 are both p/2 • Case 2.2.2.1 q1pq2 > p/2 • Induction on (p,S\{q1,q2}) •  p,p1,...,pk on S\{q1,q2} • the smaller of p1pq1 , q2pp1 is at most 3p/4 • extend to a 5p/4-open spanning path q2,q1,p,p1,...,pk or q1,q2,p,p1,...,pk on S y q1 l1 p1 T p q2 l2 z c

  23. Proof Spanning Path b b K • Case 2 (q1q2,S): K = CH(S\{q1,q2}) • Case 2.2.2.2 q1pq2p/2 • one ofq2q1p or pq2q1 is at least p/4 • q2q1b  3p/4  bpq2<p •  ypq2  bpq2<p •  q1q2py is a convex 4-gon • if q2q1p , ypq1 , yq2q1 , q2yp < 3p/4 • Induction on (py,S\{q1,q2}), and complete to a5p/4-open spanning path q2,q1,p,y,p1,...,pk or q1,q2,y,p,p1,...,pk on S y q1 l1 T w.l.o.g. pq2q1p/4 p q2 l2 z c

  24. Proof Spanning Path b K y • Case 2 (q1q2,S): K = CH(S\{q1,q2}) • Case 2.2.2.2 q1pq2p/2 • q2q1p and pq2q1 are both p/2 • yq2q1pq2q1p/2 • as the supporting line of yp intersects q1b •  pq1b + ypq1 < p  ypq1 < 3p/4 • q2pz < 3p/4 by same arguments as for ypq1 • zpy <p  ypq2  2p - p - 3p/4 = p/4 • q2yp + ypq2 <p  q2yp < 3p/4 • if q2q1p , ypq1 , yq2q1 , q2yp < 3p/4 q1 l1 T p q2 l2 z c    

  25. ??? Conclusion   • Pseudo-Triangulations: p (180°) • Perfect Matchings: 2p (360°) • Spanning Cycles: p (180°) • Triangulations: 2p/3 (120°) • Spanning Trees (unbounded): 5p/3 (300°) • Spanning Trees with bounded vertex degree: 3p/2 (270°) • Spanning Paths (convex): 3p/2 (270°) • Spanning Paths (general): 5p/4(225°) 3p/2 (270°)     

  26. Remaining questions • Closing the gap in the bounds for spanning paths • lower bound 225° — upper bound 270° • Angles of only the interior points of a pointed pseudotriangulation • simple upper bound 220° 140° 140° 40°

  27. Thanks … Thank you for your attention

  28. EWCG R A Z EWCG 07 March 19–21 2007,Graz, Austria 23rd European Workshop on Computational Geometry next week http://ewcg07.tugraz.at

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