Rectangle-of-influence triangulations Therese Biedl 1 Anna Lubiw 1 - PowerPoint PPT Presentation

Rectangle-of-influence triangulations Therese Biedl 1 Anna Lubiw 1 Saeed Mehrabi 1 Sander Verdonschot 2 1 University of Waterloo 2 University of Ottawa 5 August 2016 Sander Verdonschot Rectangle-of-influence triangulations

RI-Edges • An edge is RI if its supporting rectangle (smallest axis-aligned bounding box) is empty of (other) points Sander Verdonschot Rectangle-of-influence triangulations

RI-Drawings • Drawing of a graph where all edges are RI • Well-studied in Graph Drawing community Sander Verdonschot Rectangle-of-influence triangulations

RI-Triangulations • All internal faces are triangles • Maximal Sander Verdonschot Rectangle-of-influence triangulations

RI-Problems 1. RI-triangulating a polygon 2. RI-triangulating a point set 3. Flipping one RI-triangulation to another 4. Flipping a triangulation to an RI-triangulation Sander Verdonschot Rectangle-of-influence triangulations

RI-Polygons • All edges are RI Sander Verdonschot Rectangle-of-influence triangulations

RI-Polygons • Compute trapezoidal decomposition Sander Verdonschot Rectangle-of-influence triangulations

RI-Polygons • Add diagonal in alternating trapezoids Sander Verdonschot Rectangle-of-influence triangulations

RI-Polygons • Remaining pieces are x -monotone and one-sided Sander Verdonschot Rectangle-of-influence triangulations

RI-Polygons • Connect neighbours of local maximum Sander Verdonschot Rectangle-of-influence triangulations

RI-Polygons Theorem Every RI-polygon can be RI-triangulated in linear time. Sander Verdonschot Rectangle-of-influence triangulations

RI-Problems 1. RI-triangulating a polygon � 2. RI-triangulating a point set 3. Flipping one RI-triangulation to another 4. Flipping a triangulation to an RI-triangulation Sander Verdonschot Rectangle-of-influence triangulations

RI-Point Sets • The L ∞ -Delaunay triangulation is RI Sander Verdonschot Rectangle-of-influence triangulations

RI-Point Sets Theorem Any point set can be RI-triangulated in O ( n log n ) time. Sander Verdonschot Rectangle-of-influence triangulations

RI-Problems 1. RI-triangulating a polygon � 2. RI-triangulating a point set � 3. Flipping one RI-triangulation to another 4. Flipping a triangulation to an RI-triangulation Sander Verdonschot Rectangle-of-influence triangulations

RI-Flips • Exchange one diagonal of a convex quad for the other Sander Verdonschot Rectangle-of-influence triangulations

RI-Flips • Is the class of RI-triangulations closed under flips? • Diameter is Ω ( n 2 ) Sander Verdonschot Rectangle-of-influence triangulations

RI-Flips • Transform into the L ∞ -Delaunay triangulation • An edge is locally L ∞ if its is L ∞ w.r.t. its neighbouring triangles • If all edges are locally L ∞ , we are in the L ∞ -Delaunay triangulation Sander Verdonschot Rectangle-of-influence triangulations

RI-Flips • Flip edges that are not locally L ∞ • How do we know that new edge is (globally) RI? Sander Verdonschot Rectangle-of-influence triangulations

RI-Flips • How many flips do we need? • Give every edge a supporting square and count points inside Sander Verdonschot Rectangle-of-influence triangulations

RI-Flips • How many flips do we need? • Give every edge a supporting square and count points inside +1 Sander Verdonschot Rectangle-of-influence triangulations

RI-Flips • How many flips do we need? • Give every edge a supporting square and count points inside − 1 +1 − 1 Sander Verdonschot Rectangle-of-influence triangulations

RI-Flips Theorem The class of RI-triangulations is closed under flips and its diameter is Θ ( n 2 ) . − 1 +1 − 1 Sander Verdonschot Rectangle-of-influence triangulations

RI-Problems 1. RI-triangulating a polygon � 2. RI-triangulating a point set � 3. Flipping one RI-triangulation to another � 4. Flipping a triangulation to an RI-triangulation Sander Verdonschot Rectangle-of-influence triangulations

• We add 4 points ‘far away’ to deal with this RI-Point Sets • The outer face can be messy Sander Verdonschot Rectangle-of-influence triangulations

RI-Point Sets • The outer face can be messy • We add 4 points ‘far away’ to deal with this Sander Verdonschot Rectangle-of-influence triangulations

While getting monotonically ‘closer’? • Any triangulation can be flipped to any other in O n 2 flips [Lawson, 1972] • Some triangulations cannot be made RI in fewer than n 2 flips RI-Flips • Can we flip an arbitrary triangulation into an RI one? Sander Verdonschot Rectangle-of-influence triangulations

While getting monotonically ‘closer’? • Some triangulations cannot be made RI in fewer than n 2 flips RI-Flips • Can we flip an arbitrary triangulation into an RI one? • Any triangulation can be flipped to any other in O ( n 2 ) flips [Lawson, 1972] Sander Verdonschot Rectangle-of-influence triangulations

While getting monotonically ‘closer’? RI-Flips • Can we flip an arbitrary triangulation into an RI one? • Any triangulation can be flipped to any other in O ( n 2 ) flips [Lawson, 1972] • Some triangulations cannot be made RI in fewer than Ω ( n 2 ) flips Sander Verdonschot Rectangle-of-influence triangulations

RI-Flips • Can we flip an arbitrary triangulation into an RI one? While getting monotonically ‘closer’? • Any triangulation can be flipped to any other in O ( n 2 ) flips [Lawson, 1972] • Some triangulations cannot be made RI in fewer than Ω ( n 2 ) flips Sander Verdonschot Rectangle-of-influence triangulations

RI-Flips • Count points in ‘bad regions’ • No bad regions ⇒ the triangulation is RI Sander Verdonschot Rectangle-of-influence triangulations

Rectangle-of-influence triangulations Therese Biedl 1 Anna Lubiw 1 - PowerPoint PPT Presentation

Rectangle-of-influence triangulations Therese Biedl 1 Anna Lubiw 1 Saeed Mehrabi 1 Sander Verdonschot 2 1 University of Waterloo 2 University of Ottawa 5 August 2016 Sander Verdonschot Rectangle-of-influence triangulations RI-Edges An edge

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