Overlay Stitch Meshing 522007 Don Sheehy Overlay Stitch

Overlay Stitch Meshing 5/2/2007 Don Sheehy Overlay Stitch Meshing 1

A competitive algorithm for no-large-angle triangulation Don Sheehy Joint work with Gary Miller and Todd Phillips To appear at ICALP 2007 5/2/2007 Don Sheehy Overlay Stitch Meshing 2

The Problem Input: A Planar Straight Line Graph 5/2/2007 Don Sheehy Overlay Stitch Meshing 3

The Problem Input: A Planar Straight Line Graph Output: A Conforming Triangulation 5/2/2007 Don Sheehy Overlay Stitch Meshing 4

The Problem Input: A Planar Straight Line Graph Output: A Conforming Triangulation 5/2/2007 Don Sheehy Overlay Stitch Meshing 5

Why would you want to do that? 5/2/2007 Don Sheehy Overlay Stitch Meshing 6

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What went wrong? 5/2/2007 Don Sheehy Overlay Stitch Meshing 12

? ? What if you don’t know the function? 5/2/2007 Don Sheehy Overlay Stitch Meshing 13

2 Definitions of Quality 1. No Large Angles [Babuska, Aziz 1976] 5/2/2007 Don Sheehy Overlay Stitch Meshing 14

2 Definitions of Quality 1. No Large Angles [Babuska, Aziz 1976] 2. No Small Angles 5/2/2007 Don Sheehy Overlay Stitch Meshing 15

No Small Angles 5/2/2007 Don Sheehy Overlay Stitch Meshing 16

No Small Angles You may have heard of these before. Delaunay Refinement Sparse Voronoi Refinement Quadtree 5/2/2007 Don Sheehy Overlay Stitch Meshing 17

Paying for the spread Spread = L/s L s 5/2/2007 Don Sheehy Overlay Stitch Meshing 18

Paying for the spread Optimal No-Large-Angle Triangulation 5/2/2007 Don Sheehy Overlay Stitch Meshing 19

Paying for the spread What if we don’t allow small angles? 5/2/2007 Don Sheehy Overlay Stitch Meshing 20

Paying for the spread What if we don’t allow small angles? O(L/s) triangles! 5/2/2007 Don Sheehy Overlay Stitch Meshing 21

Paying for the spread What if we don’t allow small angles? Fact: For inputs with NO edges, no-small-angle meshing algorithms produce output with O(n log L/s) size and angles between 30 o and 120 o O(L/s) triangles! 5/2/2007 Don Sheehy Overlay Stitch Meshing 22

What to do? Small input angles can force even smaller ouput angles. [Shewchuk ’ 02] 5/2/2007 Don Sheehy Overlay Stitch Meshing 23

No Large Angles 5/2/2007 Don Sheehy Overlay Stitch Meshing 24

Polygons with Holes [Bern, Mitchell, Ruppert 95] – - All triangles are nonobtuse. - Output has O(n) triangles. 5/2/2007 Don Sheehy Overlay Stitch Meshing 25

Polygons with Holes [Bern, Mitchell, Ruppert 95] – - All triangles are nonobtuse. - Output has O(n) triangles. Does not work for arbitrary PSLGs 5/2/2007 Don Sheehy Overlay Stitch Meshing 26

Paterson’s Example Requires (n 2) points. O(n) points O(n) lines 5/2/2007 Don Sheehy Overlay Stitch Meshing 27

Paterson’s Example Requires (n 2) points. O(n) points O(n) lines 5/2/2007 Don Sheehy Overlay Stitch Meshing 28

Paterson’s Example Requires (n 2) points. O(n) points O(n) lines 5/2/2007 Don Sheehy Overlay Stitch Meshing 29

Paterson’s Example Requires (n 2) points. O(n) points O(n) lines 5/2/2007 Don Sheehy Overlay Stitch Meshing 30

Paterson’s Example Requires (n 2) points. O(n) points O(n) lines 5/2/2007 Don Sheehy Overlay Stitch Meshing 31

Paterson’s Example Requires (n 2) points. O(n) points O(n) lines 5/2/2007 Don Sheehy Overlay Stitch Meshing 32

Propagating Paths 5/2/2007 Don Sheehy Overlay Stitch Meshing 33

Propagating Paths 5/2/2007 Don Sheehy Overlay Stitch Meshing 34

Propagating Paths First introduced by Mitchell [93] Later Improved by Tan [96] Worst Case Optimal Size O(n 2) Angle bounds: 132 o 5/2/2007 Don Sheehy Overlay Stitch Meshing 35

The OSM Algorithm (Overlay Stitch Meshing) 5/2/2007 Don Sheehy Overlay Stitch Meshing 36

The OSM Algorithm (Overlay Stitch Meshing) Results: 1. Angles bounded by 170 o 2. O(log L/s) competitive size 5/2/2007 Don Sheehy Overlay Stitch Meshing 37

The OSM Algorithm (Overlay Stitch Meshing) 5/2/2007 Don Sheehy Overlay Stitch Meshing 38

The OSM Algorithm (Overlay Stitch Meshing) 5/2/2007 Don Sheehy Overlay Stitch Meshing 39

The OSM Algorithm (Overlay Stitch Meshing) 5/2/2007 Don Sheehy Overlay Stitch Meshing 40

The OSM Algorithm (Overlay Stitch Meshing) 5/2/2007 Don Sheehy Overlay Stitch Meshing 41

The OSM Algorithm (Overlay Stitch Meshing) 5/2/2007 Don Sheehy Overlay Stitch Meshing 42

The OSM Algorithm (Overlay Stitch Meshing) 5/2/2007 Don Sheehy Overlay Stitch Meshing 43

The OSM Algorithm (Overlay Stitch Meshing) 5/2/2007 Don Sheehy Overlay Stitch Meshing 44

The OSM Algorithm (Overlay Stitch Meshing) 5/2/2007 Don Sheehy Overlay Stitch Meshing 45

The OSM Algorithm (Overlay Stitch Meshing) 5/2/2007 Don Sheehy Overlay Stitch Meshing 46

The OSM Algorithm (Overlay Stitch Meshing) 5/2/2007 Don Sheehy Overlay Stitch Meshing 47

The OSM Algorithm (Overlay Stitch Meshing) 5/2/2007 Don Sheehy Overlay Stitch Meshing 48

The OSM Algorithm (Overlay Stitch Meshing) • An Overlay Edge is kept if 1. It does not intersect the input, OR 2. It forms a good intersection with the input. at least 30 o 5/2/2007 Don Sheehy Overlay Stitch Meshing 49

Angle Guarantees 5/2/2007 Don Sheehy Overlay Stitch Meshing 50

Angle Guarantees 5/2/2007 Don Sheehy Overlay Stitch Meshing 51

Angle Guarantees We want to show that angles at stitch vertices are bounded by 170 o 5/2/2007 Don Sheehy Overlay Stitch Meshing 52

Angle Guarantees An Overlay Triangle 5/2/2007 Don Sheehy Overlay Stitch Meshing 53

Angle Guarantees An Overlay Triangle An Overlay Edge 5/2/2007 Don Sheehy Overlay Stitch Meshing 54

Angle Guarantees Gap Ball An Overlay Triangle An Overlay Edge 5/2/2007 Don Sheehy Overlay Stitch Meshing 55

Angle Guarantees A Good Intersection 5/2/2007 Don Sheehy Overlay Stitch Meshing 56

Angle Guarantees A Bad Intersection 5/2/2007 Don Sheehy Overlay Stitch Meshing 57

Angle Guarantees A Bad Intersection 5/2/2007 Don Sheehy Overlay Stitch Meshing 58

Angle Guarantees How Bad can it be? A Bad Intersection 5/2/2007 Don Sheehy Overlay Stitch Meshing 59

Angle Guarantees How Bad can it be? About 10 o 5/2/2007 Don Sheehy Overlay Stitch Meshing About 170 o 60

Angle Guarantees How Bad can it be? Theorem: If any input edge makes a good intersection with an overlay edge then any other intersection on that edge is not too bad. About 10 o 5/2/2007 Don Sheehy Overlay Stitch Meshing About 170 o 61

Size Bounds 5/2/2007 Don Sheehy Overlay Stitch Meshing 62

Size Bounds How big is the resulting triangulation? 5/2/2007 Don Sheehy Overlay Stitch Meshing 63

Size Bounds How big is the resulting triangulation? Goal: log(L/s)-competitive with optimal 5/2/2007 Don Sheehy Overlay Stitch Meshing 64

Local Feature Size lfs(x) = distance to second nearest input vertex. lfs(x) 5/2/2007 x Don Sheehy Overlay Stitch Meshing Note: lfs is defined on the whole plane before we do any meshing. 65

Ruppert’s Idea r = (lfs(c)) r c Þ area(D) = (lfs(c)2) Þ # of triangles = (ss 1/lfs(x, y)2 dxdy) 5/2/2007 Don Sheehy Overlay Stitch Meshing 66

Ruppert’s Idea r = (lfs(c)) r Þ area(D) = (lfs(c)2) c Þ # of triangles = (ss 1/lfs(x, y)2 dxdy) dx This represents a (dxdy/lfs(x, y)2) fraction of the triangle. dy 5/2/2007 Don Sheehy Overlay Stitch Meshing 67

Ruppert’s Idea r = (lfs(c)) r c Þ area(D) = (lfs(c)2) Þ # of triangles = (ss lfs(x, y)-2 dxdy) Caveat: Only Works for meshes with bounded smallest angle. 5/2/2007 Don Sheehy Overlay Stitch Meshing 68

Extending Ruppert’s Idea -2 # of triangles = (ss lfs(x, y) dxdy) # of triangles along e = (sz 2 e lfs(z)-1 dz) An input edge e intersecting the overlay mesh 5/2/2007 Don Sheehy Overlay Stitch Meshing 69

Extending Ruppert’s Idea -2 # of triangles = (ss lfs(x, y) dxdy) # of triangles along e = (sz 2 e lfs(z)-1 dz) Now we can compute the size of our output mesh. We just need to compute these integrals. 5/2/2007 Don Sheehy Overlay Stitch Meshing 70

Output Size -2 Overlay Mesh Size = (ss lfs(x, y) dxdy) O(n log L/s) Stitch Vertices Added = (sz 2 E lfs(z)-1 dz) O(log L/s)£ |OPT| We’ll Prove this now 5/2/2007 Don Sheehy Overlay Stitch Meshing 71

What can we say about the optimal no-large-angle triangulation? 5/2/2007 Don Sheehy Overlay Stitch Meshing 72

Competitive Analysis Warm-up o 90 o 5/2/2007 Don Sheehy Overlay Stitch Meshing 73

Competitive Analysis Warm-up o ? o 5/2/2007 Don Sheehy Overlay Stitch Meshing 74

Competitive Analysis Warm-up 5/2/2007 Don Sheehy Overlay Stitch Meshing 75

Competitive Analysis Warm-up 5/2/2007 Don Sheehy Overlay Stitch Meshing 76

Competitive Analysis Warm-up Suppose we have a triangulation with all angles at least 170 o. the optimal 5/2/2007 Don Sheehy Overlay Stitch Meshing 77

Competitive Analysis Warm-up Suppose we have a triangulation with all angles at least 170 o. the optimal 5/2/2007 Every input edge is covered by empty lenses. Don Sheehy Overlay Stitch Meshing 78

Competitive Analysis This is what we have to integrate over. e’ We show that in each lens, we put O(log (L/s)) points on the edge. In other words: se’ 1/lfs(z) dz = O(log (L/s)) 5/2/2007 Don Sheehy Overlay Stitch Meshing 79

Competitive Analysis sz 2 e’ 1/lfs(z) dz = O(log (L/s)) e’ 5/2/2007 Don Sheehy Overlay Stitch Meshing 80

Competitive Analysis sz 2 e’ 1/lfs(z) dz = O(log (L/s)) t e’ sz 2 e’ 1/lfs(z) dz · 2 s 0 s 1/s + 2 ssl/2 1/x dx Parametrize e’ as [0, l] 1 st trick: lfs ¸ s everywhere = O(1) + O(log l/s) 2 nd trick: lfs ¸ ct for t 2 [0, l/2] 5/2/2007 Don Sheehy Overlay Stitch Meshing = O(log L/s) 81

Conclusions 5/2/2007 Don Sheehy Overlay Stitch Meshing 82

Conclusion • A new algorithm for no-large-angle triangulation. • The output has bounded degree triangles. • The first log-competitive analysis for such an algorithm. • We used some of the high school calculus you thought you forgot. 5/2/2007 Don Sheehy Overlay Stitch Meshing 83

Where to go from here? • 3 D? • Better angle bounds? • Leverage lower bound technology for constant competitive algorithm. 5/2/2007 Don Sheehy Overlay Stitch Meshing 84

Thank you. 5/2/2007 Don Sheehy Overlay Stitch Meshing 85