Download - On Nets and Meshes
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Meshes and NetsDon Sheehy
Joint work with Gary Miller and Todd Phillips
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Goals 2
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Goals 2
A new meshing algorithm.
O(n log n+m) - Optimal- Output-sensitive
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Goals 2
A new meshing algorithm.
O(n log n+m) - Optimal- Output-sensitive
A more general view of the meshing problem.
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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.
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Meshing 3Input: P ! Rd
Output: M " P with a “nice” Voronoi diagram
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Meshing 3Input: P ! Rd
Output: M " P with a “nice” Voronoi diagram
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Meshing 3Input: P ! Rd
Output: M " P with a “nice” Voronoi diagram
Why?
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Meshing 3Input: P ! Rd
Output: M " P with a “nice” Voronoi diagram
Why?- discretize space
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Meshing 3Input: P ! Rd
Output: M " P with a “nice” Voronoi diagram
Why?- discretize space- respect the input density
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Meshing 3Input: P ! Rd
Output: M " P with a “nice” Voronoi diagram
Why?- discretize space- respect the input density- applications: represent functions
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Meshing 3Input: P ! Rd
Output: M " P with a “nice” Voronoi diagram
Why?- discretize space- respect the input density- applications: represent functions- Finite Element Method
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Meshing 3Input: 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
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Overview 4
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Overview 4
Algorithm
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Overview 4
Algorithm
1 Order the inputs
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Overview 4
Algorithm
1 Order the inputs2 Incremental Updates
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Overview 4
Algorithm
1 Order the inputs2 Incremental Updates3 Refinement
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Overview 4
Algorithm
1 Order the inputs2 Incremental Updates3 Refinement4 Point Location
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Overview 4
Algorithm
1 Order the inputs2 Incremental Updates3 Refinement4 Point Location5 Finishing
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Overview 4
Algorithm
1 Order the inputs2 Incremental Updates3 Refinement4 Point Location5 Finishing
Analysis
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Overview 4
Algorithm
1 Order the inputs2 Incremental Updates3 Refinement4 Point Location5 Finishing
Analysis
1 Classic Mesh Size Analysis
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Overview 4
Algorithm
1 Order the inputs2 Incremental Updates3 Refinement4 Point Location5 Finishing
Analysis
1 Classic Mesh Size Analysis (with a twist)
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Overview 4
Algorithm
1 Order the inputs2 Incremental Updates3 Refinement4 Point Location5 Finishing
Analysis
1 Classic Mesh Size Analysis (with a twist)
2 Hierarchical Quality
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Overview 4
Algorithm
1 Order the inputs2 Incremental Updates3 Refinement4 Point Location5 Finishing
Analysis
1 Classic Mesh Size Analysis (with a twist)
2 Hierarchical Quality3 Quality => low complexity
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Overview 4
Algorithm
1 Order the inputs2 Incremental Updates3 Refinement4 Point Location5 Finishing
Analysis
1 Classic Mesh Size Analysis (with a twist)
2 Hierarchical Quality3 Quality => low complexity4 Range Space Nets
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Overview 4
Algorithm
1 Order the inputs2 Incremental Updates3 Refinement4 Point Location5 Finishing
Analysis
1 Classic Mesh Size Analysis (with a twist)
2 Hierarchical Quality3 Quality => low complexity4 Range Space Nets5 Ball Cover Lemma
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Overview 4
Algorithm
1 Order the inputs2 Incremental Updates3 Refinement4 Point Location5 Finishing
Analysis
1 Classic Mesh Size Analysis (with a twist)
2 Hierarchical Quality3 Quality => low complexity4 Range Space Nets5 Ball Cover Lemma6 Amortize Point Location Cost
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Metric Nets 5“Nets catch everything that’s big.”
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Metric Nets 5“Nets catch everything that’s big.”
Definition 1. Given a metric space X, a metric !-net
is a set N ! X such that
1. x, y " N # d(x, y) $ !
2. x " X # %y " N : d(x, y) & !
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Metric Nets 5“Nets catch everything that’s big.”
Definition 1. Given a metric space X, a metric !-net
is a set N ! X such that
1. x, y " N # d(x, y) $ !
2. x " X # %y " N : d(x, y) & !
Packing
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Metric Nets 5“Nets catch everything that’s big.”
Definition 1. Given a metric space X, a metric !-net
is a set N ! X such that
1. x, y " N # d(x, y) $ !
2. x " X # %y " N : d(x, y) & !
Packing
Covering
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Metric Nets 5“Nets catch everything that’s big.”
Definition 1. Given a metric space X, a metric !-net
is a set N ! X such that
1. x, y " N # d(x, y) $ !
2. x " X # %y " N : d(x, y) & !
Packing
Covering
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Metric Nets 5“Nets catch everything that’s big.”
Definition 1. Given a metric space X, a metric !-net
is a set N ! X such that
1. x, y " N # d(x, y) $ !
2. x " X # %y " N : d(x, y) & !
Packing
Covering
Example: Bounded Euclidean Space
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Metric Nets 5“Nets catch everything that’s big.”
Definition 1. Given a metric space X, a metric !-net
is a set N ! X such that
1. x, y " N # d(x, y) $ !
2. x " X # %y " N : d(x, y) & !
Packing
Covering
Example: Bounded Euclidean Space
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Metric Nets 5“Nets catch everything that’s big.”
Definition 1. Given a metric space X, a metric !-net
is a set N ! X such that
1. x, y " N # d(x, y) $ !
2. x " X # %y " N : d(x, y) & !
Packing
Covering
Example: Bounded Euclidean Space
Related to ball packing of radius ε/2.
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Metric Nets 5“Nets catch everything that’s big.”
Definition 1. Given a metric space X, a metric !-net
is a set N ! X such that
1. x, y " N # d(x, y) $ !
2. x " X # %y " N : d(x, y) & !
Packing
Covering
Example: Bounded Euclidean Space
Related to ball packing of radius ε/2.
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Meshes as Metric Nets 6
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Meshes as Metric Nets 6
The underlying metric is not Euclidean (but it’s close).
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Meshes as Metric Nets 6
The underlying metric is not Euclidean (but it’s close).
image source: Alice Project, INRIA
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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
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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
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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 ratioimage source: Alice Project, INRIA
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The Greedy Net Algorithm and SVR 7
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The Greedy Net Algorithm and SVR 7Add the farthest point from the current set.Repeat.
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The Greedy Net Algorithm and SVR 7Add 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
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The Greedy Net Algorithm and SVR 7Add 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]
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The Delaunay Triangulation is the dual of the Voronoi Diagram.
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The Delaunay Triangulation is the dual of the Voronoi Diagram.
8
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The Delaunay Triangulation is the dual of the Voronoi Diagram.
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The Delaunay Triangulation is the dual of the Voronoi Diagram.
8
D-ball
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Quality Meshes 9
D-Balls are Constant Ply
Total Complexity is linear in |M|
No D-ball intersects more than O(1) others.
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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 mosta constant number of times before the radius ofthe largest D-ball containing it goes down by afactor of 2.
Geometric Divide and Conquer: O(n log!)
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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 mosta constant number of times before the radius ofthe largest D-ball containing it goes down by afactor of 2.
Geometric Divide and Conquer: O(n log!)
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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 mosta constant number of times before the radius ofthe largest D-ball containing it goes down by afactor of 2.
Geometric Divide and Conquer: O(n log!)
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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 mosta constant number of times before the radius ofthe largest D-ball containing it goes down by afactor of 2.
Geometric Divide and Conquer: O(n log!)
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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 mosta constant number of times before the radius ofthe largest D-ball containing it goes down by afactor of 2.
Geometric Divide and Conquer: O(n log!)
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We replace quality with hierarchical quality. 11
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We replace quality with hierarchical quality. 11
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We replace quality with hierarchical quality. 11
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We replace quality with hierarchical quality. 11
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We replace quality with hierarchical quality. 11
Inside the cage: Old definition of quality.
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We replace quality with hierarchical quality. 11
Inside the cage: Old definition of quality.Outside: Treat the whole cage as a single object.
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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.
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With hierarchical meshes, we can add inputs in any order!
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With hierarchical meshes, we can add inputs in any order!
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With hierarchical meshes, we can add inputs in any order!
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With hierarchical meshes, we can add inputs in any order!
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Goal: Combinatorial Divide and Conquer
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With hierarchical meshes, we can add inputs in any order!
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Goal: Combinatorial Divide and Conquer
Old: Progress was decreasing the radius by a factor of 2.
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With hierarchical meshes, we can add inputs in any order!
12
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.
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Range Nets 13“Nets catch everything that’s big.”
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Range Nets 13
Definition. A range space is a pair (X,R),where X is a set (the vertices) and R is acollection of subsets (the ranges).
“Nets catch everything that’s big.”
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Range Nets 13
Definition. A range space is a pair (X,R),where X is a set (the vertices) and R is acollection of subsets (the ranges).
For us X = P and R is the set of open balls.
“Nets catch everything that’s big.”
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Range Nets 13
Definition. A range space is a pair (X,R),where X is a set (the vertices) and R is acollection of subsets (the ranges).
Definition. Given a range space (X,R),a set N ! X is a range space !-net iffor all ranges r " R that contain at least!|X| vertices, r contains a vertex from N .
For us X = P and R is the set of open balls.
“Nets catch everything that’s big.”
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Range Nets 13
Definition. A range space is a pair (X,R),where X is a set (the vertices) and R is acollection of subsets (the ranges).
Definition. Given a range space (X,R),a set N ! X is a range space !-net iffor all ranges r " R that contain at least!|X| vertices, r contains a vertex from N .
For us X = P and R is the set of open balls.
“Nets catch everything that’s big.”
Theorem: [Chazelle & Matousek 96] For ε, d fixed constants, ε-nets of size O(1) can be computed in O(n) deterministic time.
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Ordering the inputs 14For 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.
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Ordering the inputs 14For 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.
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Ordering the inputs 14For 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.
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Ordering the inputs 14For 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.
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Ordering the inputs 14For 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.
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Ordering the inputs 14For 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 Bcontains no points of M , then B is contained in theunion of d D-balls of M .
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Ordering the inputs 14For 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 Bcontains no points of M , then B is contained in theunion of d D-balls of M .
log(n) Rounds
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The D-Ball Covering Lemma 15
Lemma. Let M be a set of vertices. If an open ball Bcontains no points of M , then B is contained in theunion of d D-balls of M .
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The D-Ball Covering Lemma 15
Lemma. Let M be a set of vertices. If an open ball Bcontains no points of M , then B is contained in theunion of d D-balls of M .
Proof is constructive using Carathéodory’s Theorem.
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Amortized Cost of a Round is O(n) 16Watch an uninserted point x.Claim: x only gets touched O(1) times per round.
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Amortized Cost of a Round is O(n) 16Watch an uninserted point x.Claim: x only gets touched O(1) times per round.
x
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Amortized Cost of a Round is O(n) 16Watch an uninserted point x.Claim: x only gets touched O(1) times per round.
x
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Amortized Cost of a Round is O(n) 16Watch an uninserted point x.Claim: x only gets touched O(1) times per round.
x
y
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Amortized Cost of a Round is O(n) 16Watch an uninserted point x.Claim: x only gets touched O(1) times per round.
x
y
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Amortized Cost of a Round is O(n) 16Watch an uninserted point x.Claim: x only gets touched O(1) times per round.
y touches x => y is “close” to xclose = 2 hops among D-balls
x
y
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Amortized Cost of a Round is O(n) 16Watch an uninserted point x.Claim: x only gets touched O(1) times per round.
y touches x => y is “close” to xclose = 2 hops among D-balls
x
y
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Amortized Cost of a Round is O(n) 16Watch an uninserted point x.Claim: x only gets touched O(1) times per round.
y touches x => y is “close” to xclose = 2 hops among D-balls
Only O(1) D-balls are within 2 hops.x
y
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Amortized Cost of a Round is O(n) 16Watch an uninserted point x.Claim: x only gets touched O(1) times per round.
y touches x => y is “close” to xclose = 2 hops among D-balls
Only O(1) D-balls are within 2 hops.x
y
x
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Amortized Cost of a Round is O(n) 16Watch an uninserted point x.Claim: x only gets touched O(1) times per round.
y touches x => y is “close” to xclose = 2 hops among D-balls
Only O(1) D-balls are within 2 hops.x
y
x
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Amortized Cost of a Round is O(n) 16Watch an uninserted point x.Claim: x only gets touched O(1) times per round.
y touches x => y is “close” to xclose = 2 hops among D-balls
Only O(1) D-balls are within 2 hops.x
y
x
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Amortized Cost of a Round is O(n) 16Watch an uninserted point x.Claim: x only gets touched O(1) times per round.
y touches x => y is “close” to xclose = 2 hops among D-balls
Only O(1) D-balls are within 2 hops.x
y
Only O(1) points are added to any D-ball in a round.
x
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Amortized Cost of a Round is O(n) 16Watch an uninserted point x.Claim: x only gets touched O(1) times per round.
y touches x => y is “close” to xclose = 2 hops among D-balls
Only O(1) D-balls are within 2 hops.x
y
Only O(1) points are added to any D-ball in a round.
O(n) total work per round.
x
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Summary 17
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Summary 17Meshing
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Summary 17Meshing
Metric Nets
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Summary 17Meshing
Metric NetsGeometric D&C
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Summary 17Meshing
Metric NetsGeometric D&C
Range Nets
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Summary 17Meshing
Metric NetsGeometric D&C
Range NetsBall Covering
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Summary 17Meshing
Metric NetsGeometric D&C
Range NetsBall Covering
Combinatorial D&C
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Summary 17Meshing
Metric NetsGeometric D&C
Range NetsBall Covering
Combinatorial D&CAmortized Point Location
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Summary 17Meshing
Metric NetsGeometric D&C
Range NetsBall Covering
Combinatorial D&CAmortized Point Location
Finally, O(n log n + m) point meshing
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Summary 17Meshing
Metric NetsGeometric D&C
Range NetsBall Covering
Combinatorial D&CAmortized Point Location
Finally, O(n log n + m) point meshing
Thanks.
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Thanks again.