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

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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

- Background and motivation
- Central Idea
- Star Splaying
- Star Flipping
- Implementation
- Conclusion

Outline

- 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

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

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

- 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

- 3D convex hull -> 2D stars

Central Idea

- 2D star -> 1D link

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

- 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

- 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

- 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

- Consistency enforcement algorithm
- Reconciling an asymmetric edge
- Reconciling conflicting edge stars

A

B

C

Star Splaying -> consistency enforcement algorithm

- Consistency enforcement algorithm
- Reconciling an asymmetric edge
- Reconciling conflicting edge stars

Q

R

P

S

Star Splaying -> consistency enforcement algorithm

- 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

- 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

- 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

- 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

- Data structure

Implementation

- Making star of vertices convex

orientation determinant

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

- 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