22nd european workshop on computational geometry institute of software technology 4th fsp-seminar...

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




Plane Geometric Graphs

vertices: – n points in the plane– points in general position

edges: – straight lines spanned by vertices (geometric graphs) – no crossings (plane)

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perfect matchings





perfect matchings

spanning paths

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perfect matchings

spanning paths

spanning trees

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perfect matchings

spanning paths

spanning trees

connected plane graphs

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perfect matchings

spanning paths

spanning trees


spanning cycles

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perfect matchings

spanning paths

spanning trees


spanning cycles

triangulations

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perfect matchings

spanning paths

spanning trees


spanning cycles

triangulations

pseudo-triangulations

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Basic Idea

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Generalizing the principle of large incident angles

of pointed pseudo-triangulations to other classes of

plane graphs




Pseudo-Triangulations

pseudo-triangle– 3 convex vertices– concave chains

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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

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- 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

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p

q




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 ?

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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}

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Triangulations

convex hull points are – open

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Triangulations

convex hull points are – open

take the convex hull

triangulate

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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‘

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Triangulations

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Theorem 1: Triangulations are 2/3-open.Moreover, this bound is best possible.




Spanning Trees

not more than 5/3-open:

at least 3/2-open:

at least 5/3-open:

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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

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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

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Theorem 2: (general) Spanning Trees are 5/3-open,and this bound is best possible.




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

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Theorem 3: Spanning Trees with maximum vertex degree of at most 3 are 3/2-open.Moreover, this bound is best possible.


3/2

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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.


3/2

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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)

<

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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.

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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°)

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5/4 (225°) – 3/2 (270°)

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Thanks!

Thanks for your attention …

Grazie

Danke

Merci

GraciasEfcharisto

Dank U wel

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22nd european workshop on computational geometry institute of software technology 4th fsp-seminar...

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