Constructing the Delaunay Triangulation Algorithm 1 Randomized Incremental
- Slides: 24
Constructing the Delaunay Triangulation Algorithm 1
Randomized Incremental Construction Algorithm 2 Point
Point Addition 2) Add points in random order, maintaining a Delaunay triangulation Point of the current set.
Legal and Illegal Edges Point Replace illegal edges by legal edges through edge flips.
Edge Legalization Point An edge (which was legal before) can only become illegal if one of its two incident triangles has changed. Only the edges of the new triangles need to be checked.
More Observations Point Only the edges of the new triangles need to be checked.
Iterations Point
Correctness Need to prove that no illegal edges remain after all calls Point to Legalize. Edge. Correctness is implied by the following: Ensured by the recursive calls. Every new edge is legal. To be shown in Lemma 1 next. Any edge that may become illegal is tested. Because an edge can only become illegal if one of its incident triangles changes. Algorithm terminates because every flips increases the angle vector of the triangulation.
Legality of Every New Edge Point Proof Examine two types of edges. Case 1 Case 2
Immediately Added Edges Case 1 Point Case 2 Similar to Case 1.
Edges Added Due to Flipping The 2 nd type of edges are added due to flipping by Legalize. Edge. Point
Locating the Containing Triangle Point Trapezoidal map with only triangles no trapezoids. Internal nodes: triangles that existed before but have been destroyed.
Example
Insertion Descends to this child. Repeat the above two steps to reach a leaf. Time linear in
Example (cont’d)
Example (finish)
Point
Analysis Point Split 1 or 2 triangles, creating 3 or 4 new ones, and the same number of edges.
Proof of Lemma 2 (cont’d) 3 new edges 4 new edges 3 new triangles 4 new triangles
Triangles Generated in One Iteration 2 new triangles #new triangles
Backward Analysis Apply backward analysis to determine its expected value.
Proof of Lemma 2 (finish) Point
Storage and Run Time Point Sketch of Proof (Storage) Every node of the search structure corresponds to a triangle. (Time) Time cost is attributed to two types of operations: all point location steps
Expected Time to Locate a Point One triangle may be charged multiple times, each time for locating a different point. Total time for all point location steps is (for proof see Lemma 9. 13)
- Delaunay triangulation divide and conquer algorithm
- Delaunay triangulation divide and conquer algorithm
- Delaunay triangulation
- Bauhaus poster
- E
- Delaunay
- Pascal delaunay
- Futurism
- Expected running time of randomized algorithm
- Randomized algorithm in daa
- Incremental convex hull algorithm
- Position bresenhams line algorithm
- Triangulate
- Triangulation surveying
- Trilateration vs triangulation
- La triangulation en évaluation
- Triangulation earthquake
- What should a crime scene sketch include
- Bluetooth positioning using rssi and triangulation methods
- Triangulation vs crystallization
- Anne davies triangulation
- Triangulation police
- Triangulation meaning
- Oedipal triangulation
- Monotone polygon triangulation