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22nd European Workshop on Computational Geometry
Institute of Software Technology
4th FSP-Seminar Industrial Geometry, March 2007
Maximizing Maximal Anglesfor Plane Straight-Line GraphsO. Aichholzer, T. Hackl, M. Hoffmann, C.
Huemer, A. Pór, F. Santos, B. Speckmann, B.
Vogtenhuber
Graz University of Technology, AustriaETH Zürich, Switzerland
Universitat Politècnica de Catalunya, SpainHungarian Academy of Sciences, Hungary
Universidad de Cantabria, SpainTU Eindhoven, Netherlands
FSP-SeminarMarch 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 2007
- 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:minS maxG minpS maxaA(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
ab
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 in 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 q1q2 of the convex hull of S, there exists a 5/4-open spanning path on S starting with q1q2.
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 Matchings 2 (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
GraciasEfcharisto
Dank U wel
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