What is the difference between a mesh and a net? What is the difference between a metric space epsilon-net and a range space epsilon-net? What is the difference between geometric divide-and-conquer and combinatorial divide-and-conquer? In this talk, I will answer these questions and discuss how these different ideas come together to finally settle the question of how to compute conforming point set meshes in optimal time. The meshing problem is to discretize space into as few pieces as possible and yet still capture the underlying density of the input points. Meshes are fundamental in scientific computing, graphics, and more recently, topological data analysis. This is joint work with Gary Miller and Todd Phillips

1. Meshes and Nets
Don Sheehy
Joint work with Gary Miller and Todd Phillips

2. Goals 2

3. Goals 2
A new meshing algorithm.
O(n log n + m) - Optimal
- Output-sensitive

4. Goals 2
A new meshing algorithm.
O(n log n + m) - Optimal
- Output-sensitive
A more general view of the meshing problem.

5. Goals 2
A new meshing algorithm.
O(n log n + m) - Optimal
- Output-sensitive
A more general view of the meshing problem.
Some nice theory.

6. Meshing 3
Input: P ⊂ Rd
Output: M ⊃ P with a “nice” Voronoi diagram

7. Meshing 3
Input: P ⊂ Rd
Output: M ⊃ P with a “nice” Voronoi diagram

8. Meshing 3
Input: P ⊂ Rd
Output: M ⊃ P with a “nice” Voronoi diagram
Why?

9. Meshing 3
Input: P ⊂ Rd
Output: M ⊃ P with a “nice” Voronoi diagram
Why?
- discretize space

10. Meshing 3
Input: P ⊂ Rd
Output: M ⊃ P with a “nice” Voronoi diagram
Why?
- discretize space
- respect the input density

11. Meshing 3
Input: P ⊂ Rd
Output: M ⊃ P with a “nice” Voronoi diagram
Why?
- discretize space
- respect the input density
- applications: represent functions

12. Meshing 3
Input: P ⊂ Rd
Output: M ⊃ P with a “nice” Voronoi diagram
Why?
- discretize space
- respect the input density
- applications: represent functions
- Finite Element Method

13. Meshing 3
Input: P ⊂ Rd
Output: M ⊃ P with a “nice” Voronoi diagram
Why?
- discretize space
- respect the input density
- applications: represent functions
- Finite Element Method
- Topological Data Analysis

14. Overview 4

15. Overview 4
Algorithm

16. Overview 4
Algorithm
1 Order the inputs

17. Overview 4
Algorithm
1 Order the inputs
2 Incremental Updates

18. Overview 4
Algorithm
1 Order the inputs
2 Incremental Updates
3 Refinement

19. Overview 4
Algorithm
1 Order the inputs
2 Incremental Updates
3 Refinement
4 Point Location

20. Overview 4
Algorithm
1 Order the inputs
2 Incremental Updates
3 Refinement
4 Point Location
5 Finishing

21. Overview 4
Algorithm Analysis
1 Order the inputs
2 Incremental Updates
3 Refinement
4 Point Location
5 Finishing

22. Overview 4
Algorithm Analysis
1 Order the inputs 1 Classic Mesh Size Analysis
2 Incremental Updates
3 Refinement
4 Point Location
5 Finishing

23. Overview 4
Algorithm Analysis
1 Order the inputs 1 Classic Mesh Size Analysis
2 Incremental Updates (with a twist)
3 Refinement
4 Point Location
5 Finishing

24. Overview 4
Algorithm Analysis
1 Order the inputs 1 Classic Mesh Size Analysis
2 Incremental Updates (with a twist)
3 Refinement 2 Hierarchical Quality
4 Point Location
5 Finishing

25. Overview 4
Algorithm Analysis
1 Order the inputs 1 Classic Mesh Size Analysis
2 Incremental Updates (with a twist)
3 Refinement 2 Hierarchical Quality
4 Point Location 3 Quality => low complexity
5 Finishing

26. Overview 4
Algorithm Analysis
1 Order the inputs 1 Classic Mesh Size Analysis
2 Incremental Updates (with a twist)
3 Refinement 2 Hierarchical Quality
4 Point Location 3 Quality => low complexity
5 Finishing 4 Range Space Nets

27. Overview 4
Algorithm Analysis
1 Order the inputs 1 Classic Mesh Size Analysis
2 Incremental Updates (with a twist)
3 Refinement 2 Hierarchical Quality
4 Point Location 3 Quality => low complexity
5 Finishing 4 Range Space Nets
5 Ball Cover Lemma

28. Overview 4
Algorithm Analysis
1 Order the inputs 1 Classic Mesh Size Analysis
2 Incremental Updates (with a twist)
3 Refinement 2 Hierarchical Quality
4 Point Location 3 Quality => low complexity
5 Finishing 4 Range Space Nets
5 Ball Cover Lemma
6 Amortize Point Location Cost

29. Metric Nets 5
“Nets catch everything that’s big.”

30. Metric Nets 5
“Nets catch everything that’s big.”
Definition 1. Given a metric space X, a metric ε-net
is a set N ⊂ X such that
1. x, y ∈ N ⇒ d(x, y) ≥ ε
2. x ∈ X ⇒ ∃y ∈ N : d(x, y) ≤ ε

31. Metric Nets 5
“Nets catch everything that’s big.”
Definition 1. Given a metric space X, a metric ε-net
is a set N ⊂ X such that
1. x, y ∈ N ⇒ d(x, y) ≥ ε Packing
2. x ∈ X ⇒ ∃y ∈ N : d(x, y) ≤ ε

32. Metric Nets 5
“Nets catch everything that’s big.”
Definition 1. Given a metric space X, a metric ε-net
is a set N ⊂ X such that
1. x, y ∈ N ⇒ d(x, y) ≥ ε Packing
2. x ∈ X ⇒ ∃y ∈ N : d(x, y) ≤ ε Covering

33. Metric Nets 5
“Nets catch everything that’s big.”
Definition 1. Given a metric space X, a metric ε-net
is a set N ⊂ X such that
1. x, y ∈ N ⇒ d(x, y) ≥ ε Packing
2. x ∈ X ⇒ ∃y ∈ N : d(x, y) ≤ ε Covering

34. Metric Nets 5
“Nets catch everything that’s big.”
Definition 1. Given a metric space X, a metric ε-net
is a set N ⊂ X such that
1. x, y ∈ N ⇒ d(x, y) ≥ ε Packing
2. x ∈ X ⇒ ∃y ∈ N : d(x, y) ≤ ε Covering
Example: Bounded Euclidean Space

35. Metric Nets 5
“Nets catch everything that’s big.”
Definition 1. Given a metric space X, a metric ε-net
is a set N ⊂ X such that
1. x, y ∈ N ⇒ d(x, y) ≥ ε Packing
2. x ∈ X ⇒ ∃y ∈ N : d(x, y) ≤ ε Covering
Example: Bounded Euclidean Space

36. Metric Nets 5
“Nets catch everything that’s big.”
Definition 1. Given a metric space X, a metric ε-net
is a set N ⊂ X such that
1. x, y ∈ N ⇒ d(x, y) ≥ ε Packing
2. x ∈ X ⇒ ∃y ∈ N : d(x, y) ≤ ε Covering
Example: Bounded Euclidean Space
Related to ball packing of radius ε/2.

37. Metric Nets 5
“Nets catch everything that’s big.”
Definition 1. Given a metric space X, a metric ε-net
is a set N ⊂ X such that
1. x, y ∈ N ⇒ d(x, y) ≥ ε Packing
2. x ∈ X ⇒ ∃y ∈ N : d(x, y) ≤ ε Covering
Example: Bounded Euclidean Space
Related to ball packing of radius ε/2.

38. Meshes as Metric Nets 6

39. Meshes as Metric Nets 6
The underlying metric is not Euclidean (but it’s close).

40. Meshes as Metric Nets 6
The underlying metric is not Euclidean (but it’s close).
image source: Alice Project, INRIA

41. Meshes as Metric Nets 6
The underlying metric is not Euclidean (but it’s close).
Metric is scaled according the the “feature size”.
image source: Alice Project, INRIA

42. Meshes as Metric Nets 6
The underlying metric is not Euclidean (but it’s close).
Metric is scaled according the the “feature size”.
image source: Alice Project, INRIA

43. Meshes as Metric Nets 6
The underlying metric is not Euclidean (but it’s close).
Metric is scaled according the the “feature size”.
Quality = bounded aspect ratio
image source: Alice Project, INRIA

44. The Greedy Net Algorithm and SVR 7

45. The Greedy Net Algorithm and SVR 7
Add the farthest point from the current set.
Repeat.

46. The Greedy Net Algorithm and SVR 7
Add the farthest point from the current set.
Repeat.
Harder for infinite metric spaces.
- Figure out the metric
- Decide input or Steiner
- Do point location work

47. The Greedy Net Algorithm and SVR 7
Add the farthest point from the current set.
Repeat.
Harder for infinite metric spaces.
- Figure out the metric
- Decide input or Steiner
- Do point location work
Sparse Voronoi Refinement [Hudson, Miller, Phillips 2006]

48. The Delaunay Triangulation is the dual 8
of the Voronoi Diagram.

49. The Delaunay Triangulation is the dual 8
of the Voronoi Diagram.

50. The Delaunay Triangulation is the dual 8
of the Voronoi Diagram.

51. The Delaunay Triangulation is the dual 8
of the Voronoi Diagram.
D-ball

52. Quality Meshes 9
D-Balls are Constant Ply
Total Complexity is linear in |M|
No D-ball intersects more than O(1) others.

53. Point Location and Geometric D&C 10
Idea: Store the uninserted points in the D-balls.
When the balls change, make local updates.
Lemma (HMP06). A point is touched at most
a constant number of times before the radius of
the largest D-ball containing it goes down by a
factor of 2.
Geometric Divide and Conquer: O(n log ∆)

54. Point Location and Geometric D&C 10
Idea: Store the uninserted points in the D-balls.
When the balls change, make local updates.
Lemma (HMP06). A point is touched at most
a constant number of times before the radius of
the largest D-ball containing it goes down by a
factor of 2.
Geometric Divide and Conquer: O(n log ∆)

55. Point Location and Geometric D&C 10
Idea: Store the uninserted points in the D-balls.
When the balls change, make local updates.
Lemma (HMP06). A point is touched at most
a constant number of times before the radius of
the largest D-ball containing it goes down by a
factor of 2.
Geometric Divide and Conquer: O(n log ∆)

56. Point Location and Geometric D&C 10
Idea: Store the uninserted points in the D-balls.
When the balls change, make local updates.
Lemma (HMP06). A point is touched at most
a constant number of times before the radius of
the largest D-ball containing it goes down by a
factor of 2.
Geometric Divide and Conquer: O(n log ∆)

57. Point Location and Geometric D&C 10
Idea: Store the uninserted points in the D-balls.
When the balls change, make local updates.
Lemma (HMP06). A point is touched at most
a constant number of times before the radius of
the largest D-ball containing it goes down by a
factor of 2.
Geometric Divide and Conquer: O(n log ∆)

58. We replace quality with hierarchical quality. 11

59. We replace quality with hierarchical quality. 11

60. We replace quality with hierarchical quality. 11

61. We replace quality with hierarchical quality. 11

62. We replace quality with hierarchical quality. 11
Inside the cage: Old definition of quality.

63. We replace quality with hierarchical quality. 11
Inside the cage: Old definition of quality.
Outside: Treat the whole cage as a single object.

64. We replace quality with hierarchical quality. 11
Inside the cage: Old definition of quality.
Outside: Treat the whole cage as a single object.
All the same properties as quality meshes: ply, degree, etc.

65. With hierarchical meshes, we can add inputs 12
in any order!

66. With hierarchical meshes, we can add inputs 12
in any order!

67. With hierarchical meshes, we can add inputs 12
in any order!

68. With hierarchical meshes, we can add inputs 12
in any order!
Goal: Combinatorial Divide and Conquer

69. With hierarchical meshes, we can add inputs 12
in any order!
Goal: Combinatorial Divide and Conquer
Old: Progress was decreasing the radius by a factor of 2.

70. With hierarchical meshes, we can add inputs 12
in any order!
Goal: Combinatorial Divide and Conquer
Old: Progress was decreasing the radius by a factor of 2.
New: Decrease number of points in a ball by a factor of 2.

71. Range Nets 13
“Nets catch everything that’s big.”

72. Range Nets 13
“Nets catch everything that’s big.”
Definition. A range space is a pair (X, R),
where X is a set (the vertices) and R is a
collection of subsets (the ranges).

73. Range Nets 13
“Nets catch everything that’s big.”
Definition. A range space is a pair (X, R),
where X is a set (the vertices) and R is a
collection of subsets (the ranges).
For us X = P and R is the set of open balls.

74. Range Nets 13
“Nets catch everything that’s big.”
Definition. A range space is a pair (X, R),
where X is a set (the vertices) and R is a
collection of subsets (the ranges).
For us X = P and R is the set of open balls.
Definition. Given a range space (X, R),
a set N ⊂ X is a range space ε-net if
for all ranges r ∈ R that contain at least
ε|X| vertices, r contains a vertex from N .

75. Range Nets 13
“Nets catch everything that’s big.”
Definition. A range space is a pair (X, R),
where X is a set (the vertices) and R is a
collection of subsets (the ranges).
For us X = P and R is the set of open balls.
Definition. Given a range space (X, R),
a set N ⊂ X is a range space ε-net if
for all ranges r ∈ R that contain at least
ε|X| vertices, r contains a vertex from N .
Theorem: [Chazelle & Matousek 96] For ε, d fixed constants,
ε-nets of size O(1) can be computed in O(n) deterministic time.

76. Ordering the inputs 14
For each D-Ball, select a 1/2d-net of the points it contains.
Take the union of these nets and call it a round.
Insert these.
Repeat.

77. Ordering the inputs 14
For each D-Ball, select a 1/2d-net of the points it contains.
Take the union of these nets and call it a round.
Insert these.
Repeat.

78. Ordering the inputs 14
For each D-Ball, select a 1/2d-net of the points it contains.
Take the union of these nets and call it a round.
Insert these.
Repeat.

79. Ordering the inputs 14
For each D-Ball, select a 1/2d-net of the points it contains.
Take the union of these nets and call it a round.
Insert these.
Repeat.

80. Ordering the inputs 14
For each D-Ball, select a 1/2d-net of the points it contains.
Take the union of these nets and call it a round.
Insert these.
Repeat.

81. Ordering the inputs 14
For each D-Ball, select a 1/2d-net of the points it contains.
Take the union of these nets and call it a round.
Insert these.
Repeat.
Lemma. Let M be a set of vertices. If an open ball B
contains no points of M , then B is contained in the
union of d D-balls of M .

82. Ordering the inputs 14
For each D-Ball, select a 1/2d-net of the points it contains.
Take the union of these nets and call it a round.
Insert these.
Repeat.
log(n) Rounds
Lemma. Let M be a set of vertices. If an open ball B
contains no points of M , then B is contained in the
union of d D-balls of M .

83. The D-Ball Covering Lemma 15
Lemma. Let M be a set of vertices. If an open ball B
contains no points of M , then B is contained in the
union of d D-balls of M .

84. The D-Ball Covering Lemma 15
Lemma. Let M be a set of vertices. If an open ball B
contains no points of M , then B is contained in the
union of d D-balls of M .
Proof is constructive using Carathéodory’s Theorem.

85. Amortized Cost of a Round is O(n) 16
Watch an uninserted point x.
Claim: x only gets touched O(1) times per round.

86. Amortized Cost of a Round is O(n) 16
Watch an uninserted point x.
Claim: x only gets touched O(1) times per round.
x

87. Amortized Cost of a Round is O(n) 16
Watch an uninserted point x.
Claim: x only gets touched O(1) times per round.
x

88. Amortized Cost of a Round is O(n) 16
Watch an uninserted point x.
Claim: x only gets touched O(1) times per round.
y
x

89. Amortized Cost of a Round is O(n) 16
Watch an uninserted point x.
Claim: x only gets touched O(1) times per round.
y
x

90. Amortized Cost of a Round is O(n) 16
Watch an uninserted point x.
Claim: x only gets touched O(1) times per round.
y y touches x => y is “close” to x
close = 2 hops among D-balls
x

91. Amortized Cost of a Round is O(n) 16
Watch an uninserted point x.
Claim: x only gets touched O(1) times per round.
y y touches x => y is “close” to x
close = 2 hops among D-balls
x

92. Amortized Cost of a Round is O(n) 16
Watch an uninserted point x.
Claim: x only gets touched O(1) times per round.
y y touches x => y is “close” to x
close = 2 hops among D-balls
Only O(1) D-balls are within 2 hops.
x

93. Amortized Cost of a Round is O(n) 16
Watch an uninserted point x.
Claim: x only gets touched O(1) times per round.
y y touches x => y is “close” to x
close = 2 hops among D-balls
Only O(1) D-balls are within 2 hops.
x
x

94. Amortized Cost of a Round is O(n) 16
Watch an uninserted point x.
Claim: x only gets touched O(1) times per round.
y y touches x => y is “close” to x
close = 2 hops among D-balls
Only O(1) D-balls are within 2 hops.
x
x

95. Amortized Cost of a Round is O(n) 16
Watch an uninserted point x.
Claim: x only gets touched O(1) times per round.
y y touches x => y is “close” to x
close = 2 hops among D-balls
Only O(1) D-balls are within 2 hops.
x
x

96. Amortized Cost of a Round is O(n) 16
Watch an uninserted point x.
Claim: x only gets touched O(1) times per round.
y y touches x => y is “close” to x
close = 2 hops among D-balls
Only O(1) D-balls are within 2 hops.
x
Only O(1) points are added x
to any D-ball in a round.

97. Amortized Cost of a Round is O(n) 16
Watch an uninserted point x.
Claim: x only gets touched O(1) times per round.
y y touches x => y is “close” to x
close = 2 hops among D-balls
Only O(1) D-balls are within 2 hops.
x
Only O(1) points are added x
to any D-ball in a round.
O(n) total work per round.

98. Summary 17

99. Summary 17
Meshing

100. Summary 17
Meshing
Metric Nets

101. Summary 17
Meshing
Metric Nets
Geometric D&C

102. Summary 17
Meshing
Metric Nets
Geometric D&C
Range Nets

103. Summary 17
Meshing
Metric Nets
Geometric D&C
Range Nets
Ball Covering

104. Summary 17
Meshing
Metric Nets
Geometric D&C
Range Nets
Ball Covering
Combinatorial D&C

105. Summary 17
Meshing
Metric Nets
Geometric D&C
Range Nets
Ball Covering
Combinatorial D&C
Amortized Point Location

106. Summary 17
Meshing
Metric Nets
Geometric D&C
Range Nets
Ball Covering
Combinatorial D&C
Amortized Point Location
Finally, O(n log n + m) point meshing

107. Summary 17
Meshing
Metric Nets
Geometric D&C
Range Nets
Ball Covering
Combinatorial D&C
Amortized Point Location
Finally, O(n log n + m) point meshing
Thanks.

108. Thanks again.