Star splaying an algorithm for repairing delaunay triangulations and convex hulls
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Star Splaying: An algorithm for Repairing Delaunay Triangulations and Convex Hulls. Jonathan Richard Shewchuk. Outline. Background and motivation Central Idea Star Splaying Star Flipping Implementation Conclusion. Outline. Background and motivation. Flipping algorithm

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Star Splaying: An algorithm for Repairing Delaunay Triangulations and Convex Hulls

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Star Splaying: An algorithm for Repairing Delaunay Triangulations and Convex Hulls

Jonathan Richard Shewchuk


Outline

  • Background and motivation

  • Central Idea

  • Star Splaying

  • Star Flipping

  • Implementation

  • Conclusion

Outline


Background and motivation

  • Flipping algorithm

    • A combinatorial optimization procedure, of which the objective function maps a triangulation to a scalar value.

    • Flip a non-locally-Delaunay edge one by one, Each flipping increases the objective value.

    • Flipping is a hill climbing algorithm.

Background and motivation


Background and motivation

  • Flipping algorithm

    • E.g. In 2D triangulation, the objective value can be seen as the volume of the 3D projected polytope.

Background and motivation


Background and motivation

  • Flipping algorithm

    • In 3D or higher dimension, flipping algorithm on triangulation can get stuck in a local optimum objective value, but Delaunay.

Background and motivation


Background and motivation

  • Star splaying

    • Solve the “stuck” problem

    • Efficient when the triangulation is near to Delaunay or the polytope is nearly convex.

  • Star flipping

    • A recursive variant

Background and motivation


Central Idea

  • 3D convex hull -> 2D stars

Central Idea


Central Idea

  • 2D star -> 1D link

Central Idea


Central Idea

  • The three problems below are equivalent:

    • Computing (d+1)-dimensional convex hull H

    • Computing (d+1)-dimensional convex hull of rays HV(the cone of vertices star)

    • Computing d-dimensional convex hull P of points (intersected by a plane)

Central Idea


Central Idea

  • The idea of star splaying and star flipping

    • Every vertex maintains its own star-link structure, which is the opinion of the vertex about what the global triangulation is like.

    • The stars of different vertices may be inconsistent:

Central Idea


Central Idea

  • The idea of star splaying and star flipping

    • By iteratively communicate with neighbors, the star of a vertex splays gradually, as an umbrella.

Central Idea


Star Splaying

  • Every vertex has a link triangulation, which represents both the star and the link of the vertex.

  • Initially, every vertex links some others, with inconsistency. The initial star of a vertex may be random created, or based on the existing polytope.

  • Use consistency enforcement algorithm to fix inconsistencies, like opening umbrella.

  • When there is no inconsistency, star splaying has constructed convex hull.

Star Splaying


Star Splaying

  • Consistency enforcement algorithm

    • Reconciling an asymmetric edge

    • Reconciling conflicting edge stars

A

B

C

Star Splaying -> consistency enforcement algorithm


Star Splaying

  • Consistency enforcement algorithm

    • Reconciling an asymmetric edge

    • Reconciling conflicting edge stars

Q

R

P

S

Star Splaying -> consistency enforcement algorithm


Star Splaying

  • It is possible to get several small convex hull in a set of vertices.

  • The initial star of every vertex v contains at least one vertex that lexicographically precedes v.

Star Splaying


Star Splaying

  • for vertex v is all vertices that are on or outside the initial cone of v.

  • Thus, if the initial estimation is near the final result, star splaying consumes linear time.

Star Splaying


Star Flipping

  • Assume that we have gotten a polytope, star flipping helps to make the polytope convex.

  • Star Flipping: Making each starting cone convex by applying flipping to its link triangulation- and if that gets stuck, by calling star splaying recursively with the dimension d reduced by one.

Star Flipping


Star Flipping

  • Process of star flipping

FL

SS

DT

FL

SS

DT

FL

SS

FL

SS

FL

SS

FL

DT: Delaunay Triangulation; FL: Flipping; CEA: Consistency Enforcement Algorithm

Star Flipping


Implementation

  • Data structure

Implementation


Implementation

  • Making star of vertices convex

orientation determinant

Implementation


Implementation

  • Finding inconsistencies

    • Use a triple <f, v, w>, in which f is a facet and v, w is its vertices, to denote f is in w’s star.

    • Initially, for every vertex w, enqueue <f, v, w> for each v in the star of w.

    • When the consistency algorithm insert v into w’s star, enqueue <f, v, w>

    • Every time dequeue a <f, v, w> and use consistency algorithm on it.

Implementation


Conclusion

  • Star splaying can compute convex hull and Delaunay triangulation for arbitrary dimension.

  • Star splaying and star flipping is a good optimization tool, more suitable for repairing.

  • It supports parallel execution.

Conclusion


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