Delaunay Triangulation
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1 Delaunay Triangulation Steve Oudot slides courtesy of O. Devillers
2 MST
3 MST
4 MST use Kruskal s algorithm with Del as input O(n log n)
5 Last: lower bound for Delaunay Let x 1, x 2,..., x n R, to be sorted x i
6 Last: lower bound for Delaunay Let x 1, x 2,..., x n R, to be sorted (x 1, x 2 1),..., (x n, x 2 n) n points (x i, x 2 i ) x i
7 Last: lower bound for Delaunay Let x 1, x 2,..., x n R, to be sorted (x 1, x 2 1),..., (x n, x 2 n) n points Delaunay order in x x i
8 Last: lower bound for Delaunay Let x 1, x 2,..., x n R, to be sorted (x 1, x 2 1),..., (x n, x 2 n) n points Delaunay f(n) O(n) order in x O(n) O(n) + f(n) Ω(n log n) x i
9 Last: lower bound for Delaunay Ω(n log n)
10 Optimal algorithm for computing Delaunay Division Fusion L. J. Guibas and J. Stolfi. Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams. ACM Trans. on Graphics, 4(2):74 123, April 1985
11 Division-Fusion Classical approach example: sort Problem of size n division into 2 pbs of size O ( n /2) recursive call on sub-problems fusion
12 Division-Fusion Classical approach example: sort Problem of size n division into 2 pbs of size O ( n /2) O(n) recursive call on sub-problems 2 f ( ) n 2 fusion O(n)
13 Division-Fusion Classical approach Problem of size n fusion example: sort f(n) = O(n)+2f ( ) n 2 = O(n log n) division into 2 pbs of size O ( n /2) O(n) recursive call on sub-problems 2 f ( ) n 2 O(n)
14 Division
15 Division
16 Division
17 Division
18 Division Fusion
19 Division Fusion
20 Division Fusion
21 Division Fusion
22 Division Fusion
23 Division
24 Division
25 Division
26 Division
27 Division Fusion
28 Division Fusion
29 Division Fusion
30 Division Fusion
31 Division Fusion
32 Division
33 Division Sort in x
34 Division Sort in x store all the medians
35 Division Sort in x O(n log n) store all the medians query in O(1)
36 Fusion
37 Fusion Monochromatic triangles to be deleted
38 Fusion Bi-chromatic triangles to be construced
39
40 Constructing bi-chromatic edges from top to bottom next edge?
41 Constructing bi-chromatic edges from top to bottom next edge? rising bubble: set of circumscribed circles
42 Constructing bi-chromatic edges from top to bottom next edge? rising bubble: set of circumscribed circles
43 Constructing bi-chromatic edges from top to bottom next edge? rising bubble: set of circumscribed circles
44 Constructing bi-chromatic edges from top to bottom next edge? rising bubble: set of circumscribed circles
45 Constructing bi-chromatic edges from top to bottom next edge? rising bubble: set of circumscribed circles
46 Constructing bi-chromatic edges from top to bottom next edge? rising bubble: set of circumscribed circles
47 Constructing bi-chromatic edges from top to bottom next edge? rising bubble: set of circumscribed circles
48 Constructing bi-chromatic edges from top to bottom next edge? rising bubble: set of circumscribed circles
49 Constructing bi-chromatic edges from top to bottom next edge? rising bubble: set of circumscribed circles
50 Constructing bi-chromatic edges from top to bottom next edge? rising bubble: set of circumscribed circles
51 Constructing bi-chromatic edges from top to bottom next edge? rising bubble: set of circumscribed circles
52 Constructing bi-chromatic edges from top to bottom next edge?
53 p
54 p r b
55 first red vertex crossed by set of circles p r next
56 first blue vertex crossed by set of circles p b next
57 Only one is Delaunay p
58 first red vertex crossed by set of circles p b r r 2 r 2 circle(b, r, r 1 ) r 1 Always look at first neighbor of r ccw
59 first red vertex crossed by set of circles p r b r 2 r 3 r 3 circle(b, r, r 2 ) r 1 Always look at first neighbor of r ccw
60 first red vertex crossed by set of circles r p r 4 b r 2 r 3 r 4 circle(b, r, r 3 ) r 1 Always look at first neighbor of r ccw
61 first red vertex crossed by set of circles r 5 r p r 4 b r 2 r 3 r 5 circle(b, r, r 4 ) r 1 Always look at first neighbor of r ccw
62 first red vertex crossed by set of circles r 5 r p r 4 = r next b r 2 r 3 r 5 circle(b, r, r 4 ) r 1 red, red circle(b, r, r 4 )
63 first blue vertex crossed by set of circles p b r r next b 2 b 2 circle(b, r, b 1 ) b 1 Always look at first neighbor of b cw
64 first blue vertex crossed by set of circles p b r r next b 3 b 2 b 3 circle(b, r, b 2 ) b 1
65 first blue vertex crossed by set of circles p b r r next b 3 b next = b 2 b 3 circle(b, r, b 2 ) b 1 blue, blue circle(b, r, b 2 )
66 p b r r next b next
67 no point p r r next b b next
68 no red p r r next b b next
69 no blue p r r next b b next
70 no red point blue p r r next b b next
71 p b r r next b next
72 b p r
73 b p r r next
74 b p r b next
75 b p r
76 p b r
77 p b r r next
78 p b r b next
79 p b r
80 p b r
81 p b r r next
82 p b r
83 p b b next r
84 p b r
85
86
87
88
89 Complexity of Fusion
90 Complexity of Fusion At each step of the search for r next
91 Complexity of Fusion At each step of the search for r next A red edge is deleted
92 Complexity of Fusion At each step of the search for r next A red edge is deleted At each step of the search for b next
93 Complexity of Fusion At each step of the search for r next A red edge is deleted At each step of the search for b next A blue edge is deleted
94 Complexity of Fusion At each step of the search for r next A red edge is deleted At each step of the search for b next A blue edge is deleted After the choice between r next and b next
95 Complexity of Fusion At each step of the search for r next A red edge is deleted At each step of the search for b next A blue edge is deleted After the choice between r next and b next A black edge is created
96 Complexity of Fusion Complexity red edges + blue edges + black edges
97 Complexity of Fusion Complexity red edges + blue edges + black edges 3 n 2 + 3n 2 + 3n = O(n)
98 Overall Complexity Division = O(n) for any sub-problem + O(n log n) (preprocessing) Fusion = O(n) on sub-pb of size n Division-Fusion = O(n log n)
99 Generalizations
100 Voronoi diagram
101 Voronoi diagram Q Nearest neighbor of q among S
102 Voronoi diagram Q Nearest neighbor of q among S Change
103 Voronoi diagram Q Nearest neighbor of q among S Change ambient space (for q) IR 2 IR 3 IR d
104 Voronoi diagram Q Nearest neighbor of q among S Change metrics Euclidean L 2 L 1, L, L p hyperbolic additive weights multiplicative weights
105 Voronoi diagram Q Nearest neighbor of q among S Change universal set S points of IR d segments of IR d spheres of IR d
106 Points in IR3
107 Points in IR 3 query
108 Points in IR 3 query
109 Points in IR 3 query
110 Points in IR 3 query
111 Points in IR 3 query
112 Points in IR 3 query Nearest neighbor Voronoi diagram
113 Points in IR 3 Voronoi vertex Voronoi diagram
114 Points in IR 3 Voronoi vertex empty sphere Voronoi diagram
115 Points in IR 3 empty sphere tetrahedron Delaunay triangulation
116 Delaunay in 3D Similar to 2D: Delaunay test (empty sphere) incremental algorithm randomized algorithm duality with convex hull in 4D
117 Delaunay in 3D Similar to 2D: different from 2D variable size (linear quadratic)
118 Quadratic example
119 Quadratic example empty sphere
120 Quadratic example Ω(n 2 )
121 Size of Delaunay Θ(n 2 ) worst case Θ(n) uniformly-sampled in unit cube points O(n log n) uniformly-sampled points on a surface
122 Exotic metrics
123 Norm L : max( x, y )
124 Norm L : max( x, y ) query
125 Norm L : max( x, y ) query
126 Norm L : max( x, y ) query
127 Norm L : max( x, y ) query
128 Norm L : max( x, y ) query
129 Norm L : max( x, y ) query
130 Norm L : max( x, y ) query
131 Norm L : max( x, y ) query
132 Norm L : max( x, y ) query
133 Norm L : max( x, y ) bisector
134 Norm L : max( x, y ) bisector
135 Norm L : max( x, y )
136 Norm L : max( x, y )
137 Norm L : max( x, y )
138 Norm L : max( x, y )
139 Norm L : max( x, y ) Voronoi diagram
140 Norm L : max( x, y ) Delaunay Voronoi diagram
141 multiplicatively-weighted points 1 2
142 multiplicatively-weighted points 1 2
143 multiplicatively-weighted points 1 2
144 multiplicatively-weighted points 1 2
145 multiplicatively-weighted points 1 2
146 multiplicatively-weighted points 1 2
147 multiplicatively-weighted points 1 2
148 multiplicatively-weighted points 1 2 circular bisector
149 multiplicatively-weighted points 1 2 circular bisector
150 multiplicatively-weighted points 1 2
151 multiplicatively-weighted points 1 2
152 multiplicatively-weighted points disconnected cell 1 2
153 multiplicatively-weighted points
154 multiplicatively-weighted points
155 multiplicatively-weighted points quadratic size
156 Voronoi diagram of segments
157 Voronoi diagram of segments Q nearest segment
158 Voronoi diagram of segments parabolic bisector
159 Voronoi diagram of segments angle bisector
160 Voronoi diagram of segments points bisector
161 Voronoi diagram of segments Voronoi diagram
162 Voronoi diagram of segments
163 Voronoi diagram of segments
164 Voronoi diagram of segments Dual complex
165 That s all for today
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