Recognizing Straight Skeletons and Voronoi Diagrams and Reconstructing Their Input Therese Biedl 1 Martin Held 2 Stefan Huber 3 1 David R. Cheriton School of Computer Science University of Waterloo, Canada 2 FB Computerwissenschaften Universit¨ at Salzburg 3 Institute of Science and Technology Austria ISVD 2013 in Saint Petersburg, Russia July 8–10 Therese Biedl, Martin Held, Stefan Huber: Recognizing Straight Skeletons and Voronoi Diagrams 1 of 24
Straight skeleton of a PSLG ◮ [Aichholzer and Aurenhammer, 1998]: straight skeleton S ( G ) of a PSLG G Therese Biedl, Martin Held, Stefan Huber: Recognizing Straight Skeletons and Voronoi Diagrams Introduction / Definition of S ( H ) 2 of 24
Straight skeleton of a PSLG ◮ [Aichholzer and Aurenhammer, 1998]: straight skeleton S ( G ) of a PSLG G Therese Biedl, Martin Held, Stefan Huber: Recognizing Straight Skeletons and Voronoi Diagrams Introduction / Definition of S ( H ) 2 of 24
Straight skeleton of a PSLG ◮ [Aichholzer and Aurenhammer, 1998]: straight skeleton S ( G ) of a PSLG G Therese Biedl, Martin Held, Stefan Huber: Recognizing Straight Skeletons and Voronoi Diagrams Introduction / Definition of S ( H ) 2 of 24
Problem statement PSLG ∞ : edges may be straight-line segments or rays. Problem (GMP-SS) Given a PSLG ∞ G , can we find a PSLG H such that S ( H ) = G ? Therese Biedl, Martin Held, Stefan Huber: Recognizing Straight Skeletons and Voronoi Diagrams Introduction / Problem statement 3 of 24
Problem statement PSLG ∞ : edges may be straight-line segments or rays. Problem (GMP-SS) Given a PSLG ∞ G , can we find a PSLG H such that S ( H ) = G ? Therese Biedl, Martin Held, Stefan Huber: Recognizing Straight Skeletons and Voronoi Diagrams Introduction / Problem statement 3 of 24
Problem statement PSLG ∞ : edges may be straight-line segments or rays. Problem (GMP-SS) Given a PSLG ∞ G , can we find a PSLG H such that S ( H ) = G ? Problem [Aichholzer et al., 1995] Give necessary and sufficient conditions for G to be the straight skeleton of H . Therese Biedl, Martin Held, Stefan Huber: Recognizing Straight Skeletons and Voronoi Diagrams Introduction / Problem statement 3 of 24
Prior work [Aichholzer et al., 2012]: ◮ Any abstract tree T can be realized as S ( P ) (or V ( P )) of a convex polygon. ◮ Realizability of phylogenetic trees T as S ( P ) of a polygon P . Therese Biedl, Martin Held, Stefan Huber: Recognizing Straight Skeletons and Voronoi Diagrams Introduction / Problem statement 4 of 24
Outline Part I: Straight skeletons ◮ Characterization of straight skeletons. ◮ Three necessary and sufficient conditions for G to be the straight skeleton of a specific input. ◮ Recognizing straight skeletons. ◮ How to determine whether G is the straight skeleton of some input? ◮ Reconstruction algorithm. ◮ How to compute the input? Part II: Voronoi diagrams ◮ The framework developed in Part I can be applied to Voronoi diagrams. Therese Biedl, Martin Held, Stefan Huber: Recognizing Straight Skeletons and Voronoi Diagrams Introduction / Problem statement 5 of 24
Characterization: basic facts Facts ◮ Every edge of S ( H ) is on the bisector of two edges of H . ◮ Every face of S ( H ) contains exactly one segment of H , except for faces generated by degree-one vertices of H . ◮ Every edge of H begins and ends at an edge of S ( H ). ◮ If a vertex of S ( H ) has degree two then it coincides with a degree-one vertex of H . All other vertices have degree three or higher. Therese Biedl, Martin Held, Stefan Huber: Recognizing Straight Skeletons and Voronoi Diagrams Characterization / Three conditions 6 of 24
Characterization: basic facts Facts ◮ Every edge of S ( H ) is on the bisector of two edges of H . ◮ Every face of S ( H ) contains exactly one segment of H , except for faces generated by degree-one vertices of H . ◮ Every edge of H begins and ends at an edge of S ( H ). ◮ If a vertex of S ( H ) has degree two then it coincides with a degree-one vertex of H . All other vertices have degree three or higher. Temporary assumption: G has no degree-2 vertices. Therese Biedl, Martin Held, Stefan Huber: Recognizing Straight Skeletons and Voronoi Diagrams Characterization / Three conditions 6 of 24
Characterization: inside-condition Let G be the putative straight skeleton and F the set of faces of G . A solution to GMP-SS can be denoted as a mapping λ : F → L , where L is the set of lines. Therese Biedl, Martin Held, Stefan Huber: Recognizing Straight Skeletons and Voronoi Diagrams Characterization / Three conditions 7 of 24
Characterization: inside-condition Let G be the putative straight skeleton and F the set of faces of G . A solution to GMP-SS can be denoted as a mapping λ : F → L , where L is the set of lines. Definition (Inside-condition) λ fulfills the inside-condition if σ ( f ) := λ ( f ) ∩ f is a single line segment for all f ∈ F . Therese Biedl, Martin Held, Stefan Huber: Recognizing Straight Skeletons and Voronoi Diagrams Characterization / Three conditions 7 of 24
Characterization: inside-condition We construct H as the graph whose edges are σ ( f ), with f ∈ F . f ∈ F σ ( f ) λ ( f ) Therese Biedl, Martin Held, Stefan Huber: Recognizing Straight Skeletons and Voronoi Diagrams Characterization / Three conditions 8 of 24
Characterization: inside-condition We construct H as the graph whose edges are σ ( f ), with f ∈ F . f ∈ F σ ( f ) λ ( f ) For a G and λ we denote by G ∗ := G ∪ H and by F ∗ the faces of G ∗ . ◮ Every face of G contains two faces of G ∗ . ◮ We reuse λ ( f ) and σ ( f ) for faces of G ∗ accordingly. Therese Biedl, Martin Held, Stefan Huber: Recognizing Straight Skeletons and Voronoi Diagrams Characterization / Three conditions 8 of 24
Characterization: sweeping-condition Definition (Sweeping-condition) A face f of G ∗ fulfills the sweeping-condition if 1. f is monotone w.r.t. λ ( f ) and 2. at the lower chain, the distance to λ ( f ) is increasing , when moving away from σ ( f ). λ fulfills the sweeping-condition if all faces of G ∗ fulfill it. f ∈ F ∗ λ ( f ) Therese Biedl, Martin Held, Stefan Huber: Recognizing Straight Skeletons and Voronoi Diagrams Characterization / Three conditions 9 of 24
Characterization: bisector-condition Definition (Bisector-condition) The edge e = f ∩ f ′ fulfills the bisector-condition if e lies on the bisector of λ ( f ) and λ ( f ′ ). e f f ′ λ ( f ) λ ( f ′ ) λ fulfills the bisector-condition if all edges of G fulfill the bisector-condition. Therese Biedl, Martin Held, Stefan Huber: Recognizing Straight Skeletons and Voronoi Diagrams Characterization / Three conditions 10 of 24
Characterization Lemma If λ solves GMP-SS then λ fulfills the inside-, sweeping-, and bisector-condition. Inside- and bisector-condition: by definition of straight skeletons. Proof. Sweeping-condition: ◮ Monotonicity by [Aichholzer et al., 1995]. ◮ Lower chain is even convex by [Huber, 2012]. Therese Biedl, Martin Held, Stefan Huber: Recognizing Straight Skeletons and Voronoi Diagrams Characterization / Main theorem 11 of 24
Characterization Lemma If λ solves GMP-SS then λ fulfills the inside-, sweeping-, and bisector-condition. Inside- and bisector-condition: by definition of straight skeletons. Proof. Sweeping-condition: ◮ Monotonicity by [Aichholzer et al., 1995]. ◮ Lower chain is even convex by [Huber, 2012]. Theorem If λ fulfills the inside-, sweeping-, and bisector-condition then λ solves GMP-SS. Therese Biedl, Martin Held, Stefan Huber: Recognizing Straight Skeletons and Voronoi Diagrams Characterization / Main theorem 11 of 24
Recognizing straight skeletons Key method: We successively reflect lines λ ( f ) at edges of f . ◮ Assume we know a suitable λ ( f ) for one face f . ◮ Bisector-condition: we know λ ( f ′ ) for a neighboring face f ′ , too. ◮ Going along a spanning tree of the dual of G , we know λ ( f ′ ) for all f ′ ∈ F . e f f ′ λ ( f ) λ ( f ′ ) Therese Biedl, Martin Held, Stefan Huber: Recognizing Straight Skeletons and Voronoi Diagrams Recognizing S ( G ) / Star graph 12 of 24
Recognizing straight skeletons Key method: We successively reflect lines λ ( f ) at edges of f . ◮ Assume we know a suitable λ ( f ) for one face f . ◮ Bisector-condition: we know λ ( f ′ ) for a neighboring face f ′ , too. ◮ Going along a spanning tree of the dual of G , we know λ ( f ′ ) for all f ′ ∈ F . Φ e 1 ◦ e 2 ( l ) = Φ e 2 (Φ e 1 ( l )) e 2 e 1 point set l Φ − 1 e 1 ◦ e 2 = Φ e 2 ◦ e 1 Φ e 1 ( l ) Therese Biedl, Martin Held, Stefan Huber: Recognizing Straight Skeletons and Voronoi Diagrams Recognizing S ( G ) / Star graph 12 of 24
Recognizing straight skeletons: star graphs ◮ “Local view” at a vertex v of G with incident ray-edges b 1 , . . . , b d . ◮ Find λ that fulfills inside-, (sweeping-), and bisector-condition. b 2 f 2 f 1 b 1 b 3 v b 4 f d b d Therese Biedl, Martin Held, Stefan Huber: Recognizing Straight Skeletons and Voronoi Diagrams Recognizing S ( G ) / Star graph 13 of 24
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