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5/2/2007 Don SheehyOverlay Stitch Meshing
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A competitive algorithm for A competitive algorithm for no-large-angle triangulationno-large-angle triangulation
Don SheehyDon Sheehy
Joint work with Gary Miller Joint work with Gary Miller and Todd Phillipsand Todd Phillips
To appear at ICALP 2007To appear at ICALP 2007
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The ProblemThe ProblemInput: A Planar Straight Line Input: A Planar Straight Line
GraphGraph
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The ProblemThe ProblemInput: A Planar Straight Line Input: A Planar Straight Line
GraphGraph
Output: A Conforming TriangulationOutput: A Conforming Triangulation
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The ProblemThe ProblemInput: A Planar Straight Line Input: A Planar Straight Line
GraphGraph
Output: A Conforming TriangulationOutput: A Conforming Triangulation
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Why would you want to do that?Why would you want to do that?
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Why would you want to do that?Why would you want to do that?
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Why would you want to do that?Why would you want to do that?
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Why would you want to do that?Why would you want to do that?
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Why would you want to do that?Why would you want to do that?
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Why would you want to do that?Why would you want to do that?
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What if you don’t know the function?What if you don’t know the function?
?? ??
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2 Definitions of Quality2 Definitions of Quality
1. No Large Angles [Babuska, Aziz 1. No Large Angles [Babuska, Aziz 1976]1976]
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2 Definitions of Quality2 Definitions of Quality
1. No Large Angles [Babuska, Aziz 1. No Large Angles [Babuska, Aziz 1976]1976]
2. No Small Angles2. No Small Angles
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No Small AnglesNo Small Angles
You may have heard of these You may have heard of these before.before.
Delaunay RefinementDelaunay Refinement
Sparse Voronoi RefinementSparse Voronoi Refinement
QuadtreeQuadtree
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Paying for the spreadPaying for the spread
LL
ss
Spread = L/sSpread = L/s
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Paying for the spreadPaying for the spread
Optimal No-Large-Angle TriangulationOptimal No-Large-Angle Triangulation
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Paying for the spreadPaying for the spreadWhat if we don’t allow small angles?What if we don’t allow small angles?
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Paying for the spreadPaying for the spreadWhat if we don’t allow small angles?What if we don’t allow small angles?
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Paying for the spreadPaying for the spread
O(L/s)O(L/s) triangles! triangles!
What if we don’t allow small angles?What if we don’t allow small angles?
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Paying for the spreadPaying for the spread
O(L/s)O(L/s) triangles! triangles!
What if we don’t allow small angles?What if we don’t allow small angles?Fact: For inputs with Fact: For inputs with NO NO edgesedges, no-small-angle , no-small-angle
meshing algorithms produce meshing algorithms produce output with output with O(n log L/s)O(n log L/s) size size and angles between 30and angles between 30oo and and
120120oo
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What to do?What to do?
Small input angles can force even smaller Small input angles can force even smaller ouput angles. [Shewchuk ’02]ouput angles. [Shewchuk ’02]
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Polygons with HolesPolygons with Holes[Bern, Mitchell, Ruppert 95] – [Bern, Mitchell, Ruppert 95] –
- All triangles are - All triangles are nonobtuse. nonobtuse.
- Output has O(n) triangles.- Output has O(n) triangles.
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Polygons with HolesPolygons with Holes[Bern, Mitchell, Ruppert 95] – [Bern, Mitchell, Ruppert 95] –
- All triangles are - All triangles are nonobtuse. nonobtuse.
- Output has O(n) triangles.- Output has O(n) triangles.
Does not work for arbitrary PSLGsDoes not work for arbitrary PSLGs
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Paterson’s ExamplePaterson’s ExampleRequires Requires (n(n22) points.) points.
O(n) O(n) pointpoint
ss
O(n) O(n) lineline
ss
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Paterson’s ExamplePaterson’s ExampleRequires Requires (n(n22) points.) points.
O(n) O(n) pointpoint
ss
O(n) O(n) lineline
ss
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Paterson’s ExamplePaterson’s ExampleRequires Requires (n(n22) points.) points.
O(n) O(n) pointpoint
ss
O(n) O(n) lineline
ss
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Paterson’s ExamplePaterson’s ExampleRequires Requires (n(n22) points.) points.
O(n) O(n) pointpoint
ss
O(n) O(n) lineline
ss
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Paterson’s ExamplePaterson’s ExampleRequires Requires (n(n22) points.) points.
O(n) O(n) pointpoint
ss
O(n) O(n) lineline
ss
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Paterson’s ExamplePaterson’s ExampleRequires Requires (n(n22) points.) points.
O(n) O(n) pointpoint
ss
O(n) O(n) lineline
ss
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Propagating PathsPropagating Paths
First introduced by Mitchell [93]First introduced by Mitchell [93]
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Propagating PathsPropagating Paths
First introduced by Mitchell [93]First introduced by Mitchell [93]
Later Improved by Tan [96]Later Improved by Tan [96]
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Propagating PathsPropagating Paths
First introduced by Mitchell [93]First introduced by Mitchell [93]
Later Improved by Tan [96]Later Improved by Tan [96]
Worst Case Optimal Size O(nWorst Case Optimal Size O(n22))
Angle bounds: 132Angle bounds: 132oo
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The OSM AlgorithmThe OSM Algorithm(Overlay Stitch Meshing)(Overlay Stitch Meshing)
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The OSM AlgorithmThe OSM Algorithm(Overlay Stitch Meshing)(Overlay Stitch Meshing)
Results:Results:1.1. Angles bounded by 170Angles bounded by 170oo
2.2. O(log L/s) competitive sizeO(log L/s) competitive size
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The OSM AlgorithmThe OSM Algorithm(Overlay Stitch Meshing)(Overlay Stitch Meshing)
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The OSM AlgorithmThe OSM Algorithm(Overlay Stitch Meshing)(Overlay Stitch Meshing)
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The OSM AlgorithmThe OSM Algorithm(Overlay Stitch Meshing)(Overlay Stitch Meshing)
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The OSM AlgorithmThe OSM Algorithm(Overlay Stitch Meshing)(Overlay Stitch Meshing)
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The OSM AlgorithmThe OSM Algorithm(Overlay Stitch Meshing)(Overlay Stitch Meshing)
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The OSM AlgorithmThe OSM Algorithm(Overlay Stitch Meshing)(Overlay Stitch Meshing)
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The OSM AlgorithmThe OSM Algorithm(Overlay Stitch Meshing)(Overlay Stitch Meshing)
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The OSM AlgorithmThe OSM Algorithm(Overlay Stitch Meshing)(Overlay Stitch Meshing)
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The OSM AlgorithmThe OSM Algorithm(Overlay Stitch Meshing)(Overlay Stitch Meshing)
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The OSM AlgorithmThe OSM Algorithm(Overlay Stitch Meshing)(Overlay Stitch Meshing)
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The OSM AlgorithmThe OSM Algorithm(Overlay Stitch Meshing)(Overlay Stitch Meshing)
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The OSM AlgorithmThe OSM Algorithm(Overlay Stitch Meshing)(Overlay Stitch Meshing)
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The OSM AlgorithmThe OSM Algorithm(Overlay Stitch Meshing)(Overlay Stitch Meshing)
• An Overlay Edge is An Overlay Edge is keptkept if if1. It does not intersect the input,1. It does not intersect the input,
OROR
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The OSM AlgorithmThe OSM Algorithm(Overlay Stitch Meshing)(Overlay Stitch Meshing)
• An Overlay Edge is An Overlay Edge is keptkept if if1. It does not intersect the input,1. It does not intersect the input,
OROR
2. It forms a 2. It forms a goodgood intersection with the intersection with the input.input.
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The OSM AlgorithmThe OSM Algorithm(Overlay Stitch Meshing)(Overlay Stitch Meshing)
• An Overlay Edge is An Overlay Edge is keptkept if if1. It does not intersect the input,1. It does not intersect the input,
OROR
2. It forms a 2. It forms a goodgood intersection with the intersection with the input.input.
at least 30at least 30oo
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Angle GuaranteesAngle Guarantees
We want to show that angles at We want to show that angles at stitchstitch vertices are bounded by vertices are bounded by
170170oo
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Angle GuaranteesAngle Guarantees
An Overlay TriangleAn Overlay Triangle
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Angle GuaranteesAngle Guarantees
An Overlay TriangleAn Overlay Triangle
An Overlay EdgeAn Overlay Edge
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Angle GuaranteesAngle Guarantees
An Overlay TriangleAn Overlay Triangle
An Overlay EdgeAn Overlay Edge
Gap BallGap Ball
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Angle GuaranteesAngle Guarantees
A Good IntersectionA Good Intersection
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Angle GuaranteesAngle Guarantees
A Bad IntersectionA Bad Intersection
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Angle GuaranteesAngle Guarantees
A Bad IntersectionA Bad Intersection
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Angle GuaranteesAngle Guarantees
A Bad IntersectionA Bad Intersection
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Angle GuaranteesAngle Guarantees
A Bad IntersectionA Bad Intersection
How Bad can it How Bad can it be?be?
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Angle GuaranteesAngle GuaranteesHow Bad can it How Bad can it
be?be?
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Angle GuaranteesAngle GuaranteesHow Bad can it How Bad can it
be?be?
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Angle GuaranteesAngle GuaranteesHow Bad can it How Bad can it
be?be?
About 10About 10oo
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Angle GuaranteesAngle GuaranteesHow Bad can it How Bad can it
be?be?
About 10About 10oo
About 170About 170oo
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Angle GuaranteesAngle GuaranteesHow Bad can it How Bad can it
be?be?
About 10About 10oo
Theorem: If any input Theorem: If any input edge makes a edge makes a goodgood intersection with an intersection with an overlay edge then any overlay edge then any other intersection on other intersection on that edge is that edge is not too not too badbad..
About 170About 170oo
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Size BoundsSize BoundsHow big is the resulting triangulation?How big is the resulting triangulation?
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Size BoundsSize BoundsHow big is the resulting triangulation?How big is the resulting triangulation?
Goal: log(L/s)-competitive with optimalGoal: log(L/s)-competitive with optimal
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Local Feature SizeLocal Feature Size
lfs(x) = distance to second nearest input vertex.lfs(x) = distance to second nearest input vertex.
xxlfs(x)lfs(x)
Note: lfs is defined on Note: lfs is defined on the whole plane before the whole plane before
we do any meshing.we do any meshing.
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Ruppert’s IdeaRuppert’s Idea
rr
cc
r = r = (lfs(c))(lfs(c))
area(area() = ) = (lfs(c)(lfs(c)22))
# of triangles = # of triangles = ((ssss 1/lfs(x,y) 1/lfs(x,y)22 dxdy) dxdy)
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Ruppert’s IdeaRuppert’s Idea
rr
cc
r = r = (lfs(c))(lfs(c))
area(area() = ) = (lfs(c)(lfs(c)22))
# of triangles = # of triangles = ((ssss 1/lfs(x,y) 1/lfs(x,y)22 dxdy) dxdy)
dydy
dxdxThis represents a This represents a (dxdy/lfs(x,y)(dxdy/lfs(x,y)22)) fraction of the fraction of the
triangle.triangle.
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Ruppert’s IdeaRuppert’s Idea
rr
cc
r = r = (lfs(c))(lfs(c))
area(area() = ) = (lfs(c)(lfs(c)22))
# of triangles = # of triangles = ((ssss lfs(x,y) lfs(x,y)-2-2 dxdy) dxdy)
CaveatCaveat: Only Works for meshes : Only Works for meshes
with bounded with bounded smallestsmallest angle. angle.
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ExtendingExtending Ruppert’s Idea Ruppert’s Idea# of triangles = # of triangles = ((ssss lfs(x,y) lfs(x,y)-2-2 dxdy) dxdy)
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ExtendingExtending Ruppert’s Idea Ruppert’s Idea
An input edge An input edge e e intersecting intersecting
the overlay meshthe overlay mesh
# of triangles = # of triangles = ((ssss lfs(x,y) lfs(x,y)-2-2 dxdy) dxdy)
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ExtendingExtending Ruppert’s Idea Ruppert’s Idea
An input edge An input edge e e intersecting intersecting
the overlay meshthe overlay mesh
# of triangles along # of triangles along ee = = ((sszz22 e e lfs(z) lfs(z)-1-1 dz) dz)
# of triangles = # of triangles = ((ssss lfs(x,y) lfs(x,y)-2-2 dxdy) dxdy)
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ExtendingExtending Ruppert’s Idea Ruppert’s Idea
Now we can compute the Now we can compute the size of our output mesh. We size of our output mesh. We just need to compute these just need to compute these
integrals.integrals.
# of triangles along # of triangles along ee = = ((sszz22 e e lfs(z) lfs(z)-1-1 dz) dz)
# of triangles = # of triangles = ((ssss lfs(x,y) lfs(x,y)-2-2 dxdy) dxdy)
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Output SizeOutput Size
Stitch Vertices Added = Stitch Vertices Added = ((sszz22 E E lfs(z) lfs(z)-1-1 dz) dz)
Overlay Mesh Size = Overlay Mesh Size = ((ssss lfs(x,y) lfs(x,y)-2-2 dxdy) dxdy)
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Output SizeOutput Size
Stitch Vertices Added = Stitch Vertices Added = ((sszz22 E E lfs(z) lfs(z)-1-1 dz) dz)
Overlay Mesh Size = Overlay Mesh Size = ((ssss lfs(x,y) lfs(x,y)-2-2 dxdy) dxdy) O(n log L/s)O(n log L/s)
O(log L/s)O(log L/s)££ |OPT| |OPT|
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Output SizeOutput Size
Stitch Vertices Added = Stitch Vertices Added = ((sszz22 E E lfs(z) lfs(z)-1-1 dz) dz)
Overlay Mesh Size = Overlay Mesh Size = ((ssss lfs(x,y) lfs(x,y)-2-2 dxdy) dxdy) O(n log L/s)O(n log L/s)
O(log L/s)O(log L/s)££ |OPT| |OPT|
We’ll Prove this nowWe’ll Prove this now
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What can we say about the What can we say about the optimal optimal no-large-angle no-large-angle
triangulation?triangulation?
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Competitive AnalysisCompetitive AnalysisWarm-upWarm-up
9090oo
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Competitive AnalysisCompetitive AnalysisWarm-upWarm-up
??oo
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Competitive AnalysisCompetitive AnalysisWarm-upWarm-up
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Competitive AnalysisCompetitive AnalysisWarm-upWarm-up
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Competitive AnalysisCompetitive AnalysisWarm-upWarm-up
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Competitive AnalysisCompetitive AnalysisWarm-upWarm-up
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Competitive AnalysisCompetitive AnalysisWarm-upWarm-up
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Competitive AnalysisCompetitive AnalysisWarm-upWarm-up
Suppose we have a triangulation with all angles at least 170Suppose we have a triangulation with all angles at least 170oo..
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Competitive AnalysisCompetitive AnalysisWarm-upWarm-up
Suppose we have a triangulation with all angles at least 170Suppose we have a triangulation with all angles at least 170oo..
the optimalthe optimal
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Competitive AnalysisCompetitive AnalysisWarm-upWarm-up
Suppose we have a triangulation with all angles at least 170Suppose we have a triangulation with all angles at least 170oo..
the optimalthe optimal Every input edge is Every input edge is covered by empty covered by empty
lenses.lenses.
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Competitive AnalysisCompetitive Analysis
Suppose we have a triangulation with all angles at least 170Suppose we have a triangulation with all angles at least 170oo..
the optimalthe optimal Every input edge is Every input edge is covered by empty covered by empty
lenses.lenses.
This is what we have to integrate over.This is what we have to integrate over.
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Competitive AnalysisCompetitive AnalysisThis is what we have to integrate over.This is what we have to integrate over.
We show that in each lens, we We show that in each lens, we put O(log (L/s)) points on the put O(log (L/s)) points on the
edge.edge.
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Competitive AnalysisCompetitive AnalysisThis is what we have to integrate over.This is what we have to integrate over.
We show that in each lens, we We show that in each lens, we put O(log (L/s)) points on the put O(log (L/s)) points on the
edge.edge.
In other words:In other words:
sse’e’ 1/lfs(z) dz = O(log (L/s)) 1/lfs(z) dz = O(log (L/s))
ee’’
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Competitive AnalysisCompetitive Analysis
ee’’
sszz22 e’ e’ 1/lfs(z) dz = O(log (L/s)) 1/lfs(z) dz = O(log (L/s))
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Competitive AnalysisCompetitive Analysis
ee’’
Parametrize e’ as [0, Parametrize e’ as [0, ll] where ] where ll = length(e’) = length(e’)
sszz22 e’ e’ 1/lfs(z) dz = O(log (L/s)) 1/lfs(z) dz = O(log (L/s))
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Competitive AnalysisCompetitive Analysis
ee’’
sszz22 e’ e’ 1/lfs(z) dz = O(log (L/s)) 1/lfs(z) dz = O(log (L/s))
Parametrize e’ as [0, Parametrize e’ as [0, ll]]
11stst trick: lfs trick: lfs ¸̧ s everywhere s everywhere
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Competitive AnalysisCompetitive Analysissszz22 e’ e’ 1/lfs(z) dz = O(log (L/s)) 1/lfs(z) dz = O(log (L/s))
ee’’
11stst trick: lfs trick: lfs ¸̧ s everywhere s everywhere
22ndnd trick: lfs trick: lfs ¸̧ ct for t ct for t22 [0, [0,ll/2]/2]
tt
Parametrize e’ as [0, Parametrize e’ as [0, ll]]
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Competitive AnalysisCompetitive Analysissszz22 e’ e’ 1/lfs(z) dz = O(log (L/s)) 1/lfs(z) dz = O(log (L/s))
ee’’
11stst trick: lfs trick: lfs ¸̧ s everywhere s everywhere
22ndnd trick: lfs trick: lfs ¸̧ ct for ct for tt22 [0, [0,ll/2]/2]
tt
sszz22 e’ e’ 1/lfs(z) dz 1/lfs(z) dz ·· 2 2ss00ss 1/s + 2 1/s + 2ssss
ll/2/2 1/x 1/x dx dx
= O(1) + O(log = O(1) + O(log ll/s)/s)
= O(log L/s)= O(log L/s)
Parametrize e’ as [0, Parametrize e’ as [0, ll]]
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ConclusionConclusion• A new algorithm for no-large-angle A new algorithm for no-large-angle
triangulation.triangulation.• The output has bounded degree The output has bounded degree
triangles.triangles.• The first log-competitive analysis for The first log-competitive analysis for
such an algorithm.such an algorithm.
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ConclusionConclusion• A new algorithm for no-large-angle A new algorithm for no-large-angle
triangulation.triangulation.• The output has bounded degree The output has bounded degree
triangles.triangles.• The first log-competitive analysis for The first log-competitive analysis for
such an algorithm.such an algorithm.• We used some of the high school We used some of the high school
calculus you thought you forgot.calculus you thought you forgot.
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Where to go from here?Where to go from here?• 3D?3D?• Better angle bounds?Better angle bounds?
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Where to go from here?Where to go from here?• 3D?3D?• Better angle bounds?Better angle bounds?• Leverage lower bound technology for Leverage lower bound technology for
constant competitive algorithm.constant competitive algorithm.