Flips in Edge-Labelled Pseudo-Triangulations Prosenjit Bose Sander - PowerPoint PPT Presentation

Flips in Edge-Labelled Pseudo-Triangulations Prosenjit Bose Sander Verdonschot Carleton University 10 August 2015 Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

• Pointed: all vertices are incident to a reflex angle ( ) Pseudo-triangulations • All faces are pseudo-triangles Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

Pseudo-triangulations • All faces are pseudo-triangles • Pointed: all vertices are incident to a reflex angle ( > π ) Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

Flips • Remove edge, leaving a pseudo-quadrilateral • Find corners opposite removed edge • Insert connecting geodesic Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

• What happens when edges are labelled? Previous work . Theorem (Bereg, 2004) . Any pointed pseudo-triangulation can be transformed into any other with O ( n log n ) flips. . Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

Previous work . Theorem (Bereg, 2004) . Any pointed pseudo-triangulation can be transformed into any other with O ( n log n ) flips. . • What happens when edges are labelled? Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

• Via canonical form T C • We only need to show T T C Upper bound • Transform T 1 into T 2 Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

• We only need to show T T C Upper bound • Transform T 1 into T 2 • Via canonical form T C Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

Upper bound • Transform T 1 into T 2 • Via canonical form T C • We only need to show T �→ T C Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

Transform into canonical • Ignore labels • Move labels around Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

Left-shelling pseudo-triangulation • Add vertices in clockwise order around bottom vertex • Connect to bottom (bottom edge) • Add tangent to convex hull (top edge) Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

Tools • Sweep : exchange labels on top and bottom pairs • Shuffle : reorder bottom labels Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

Algorithm • Identify out-of-place top and bottom labels Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

Algorithm • Pair these up ( Shuffle ) Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

Algorithm • Exchange them ( Sweep ) Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

Algorithm • Sort bottom labels ( Shuffle ) Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

Algorithm • Move all top labels down ( Sweep ) Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

Algorithm • Sort them ( Shuffle ) Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

Algorithm • Move them back ( Sweep ) Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

• How do we shuffle and sweep? Upper bound . Theorem . We can sort the labels of a left-shelling pseudo-triangulation with O ( 1 ) shuffles and sweeps. . Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

Upper bound . Theorem . We can sort the labels of a left-shelling pseudo-triangulation with O ( 1 ) shuffles and sweeps. . • How do we shuffle and sweep? Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

Sweep • Easy for degree-2 vertices: • Idea: make every vertex degree-2 at some point Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

When it passes a vertex: • Swap the top and bottom edge, if necessary • Flip the top edge Sweep • Shoot a ray from v bottom to the right Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

When it passes a vertex: • Swap the top and bottom edge, if necessary • Flip the top edge Sweep • Sweep it counter-clockwise through the point set Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

Sweep • When it passes a vertex: • Swap the top and bottom edge, if necessary • Flip the top edge Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

Tools • Sweep : exchange labels on top and bottom pairs — O ( n ) • Shuffle : reorder bottom labels Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

• Otherwise, flip top edge first • We can do insertion sort! Shuffle • Swap consecutive bottom edges • Easy if third pseudo-triangle is a triangle Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

• We can do insertion sort! Shuffle • Swap consecutive bottom edges • Easy if third pseudo-triangle is a triangle • Otherwise, flip top edge first Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

Shuffle • Swap consecutive bottom edges • Easy if third pseudo-triangle is a triangle • Otherwise, flip top edge first • We can do insertion sort! Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

Flips in Edge-Labelled Pseudo-Triangulations Prosenjit Bose Sander - PowerPoint PPT Presentation

Flips in Edge-Labelled Pseudo-Triangulations Prosenjit Bose Sander Verdonschot Carleton University 10 August 2015 Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations Pointed: all vertices are incident to a

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