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![Page 1: Tools from Computational Geometry Bernard Chazelle Princeton University Bernard Chazelle Princeton University Tutorial FOCS 2005.](https://reader037.fdocuments.in/reader037/viewer/2022103022/56649cef5503460f949bdb3c/html5/thumbnails/1.jpg)
Tools from Computational Tools from Computational GeometryGeometry
Tools from Computational Tools from Computational GeometryGeometry
Bernard ChazelleBernard Chazelle
Princeton UniversityPrinceton University
Bernard ChazelleBernard Chazelle
Princeton UniversityPrinceton University
Tutorial FOCS 2005Tutorial FOCS 2005Tutorial FOCS 2005Tutorial FOCS 2005
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Tools from Computational Tools from Computational GeometryGeometry
Tools from Computational Tools from Computational GeometryGeometry
Bernard ChazelleBernard Chazelle
Princeton UniversityPrinceton University
Bernard ChazelleBernard Chazelle
Princeton UniversityPrinceton University
Tutorial FOCS 1905Tutorial FOCS 1905Tutorial FOCS 1905Tutorial FOCS 1905
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Ruler & Compass AlgorithmsRuler & Compass AlgorithmsRuler & Compass AlgorithmsRuler & Compass Algorithms
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Gauss: 17-gonGauss: 17-gonGauss: 17-gonGauss: 17-gon
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Gauss: 17-gonGauss: 17-gonGauss: 17-gonGauss: 17-gon
)17/sin(
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Constructing Regular N-gonsConstructing Regular N-gonsConstructing Regular N-gonsConstructing Regular N-gons
33 folklorefolklore
55 antiquityantiquity
1717 Gauss (1796)Gauss (1796)
257257 Richelot (1832)Richelot (1832)
6553765537 Hermes (1879)Hermes (1879)
Gauss Gauss Fermat primes 2 Fermat primes 2
22 kk
can’t do can’t do heptagonsheptagons
+1 +1
proof coversproof coversa gyma gym
Hilbert proved lower bounds on number of stepsHilbert proved lower bounds on number of steps
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Tools from Computational Tools from Computational GeometryGeometry
Tools from Computational Tools from Computational GeometryGeometry
Bernard ChazelleBernard Chazelle
Princeton UniversityPrinceton University
Bernard ChazelleBernard Chazelle
Princeton UniversityPrinceton University
Tutorial FOCS 2005Tutorial FOCS 2005Tutorial FOCS 2005Tutorial FOCS 2005
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algorithmicalgorithmic algorithmicalgorithmic
analyticalanalytical analyticalanalytical
TOOLS FROM COMPUTATIONAL GEOMETRYTOOLS FROM COMPUTATIONAL GEOMETRY TOOLS FROM COMPUTATIONAL GEOMETRYTOOLS FROM COMPUTATIONAL GEOMETRY
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1 Algorithmic tools1 Algorithmic tools 1 Algorithmic tools1 Algorithmic tools
geometric divide & geometric divide & conquerconquer
geometric divide & geometric divide & conquerconquer
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Voronoi Diagram
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Voronoi Diagram
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Works well also for convex hulls, nearest neighbors [3,6]
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Works not so well for multidimensional searching:
quadtrees, kd-trees: highly sub-optimal
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Hopcroft’s problemHopcroft’s problemAny point/line incidence?N points and N lines
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Naïve divide & conquer
N
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O(N logN) time
N
Point location in line arrangement
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~O(N ) time
3/2
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1 Algorithmic tools1 Algorithmic tools 1 Algorithmic tools1 Algorithmic tools
geometric divide & geometric divide & conquerconquer
geometric divide & geometric divide & conquerconquer
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1 Algorithmic tools1 Algorithmic tools 1 Algorithmic tools1 Algorithmic tools
geometric divide & geometric divide & conquerconquer
geometric divide & geometric divide & conquerconquer
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[2, p.123]
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N points
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N points number of intersections = O( )
N
for any line
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Often, the number of simple polygons is exponential.
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Sometimes, it’s unique…
Often, the number of simple polygons is exponential.
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Sometimes, it’s unique…
Often, the number of simple polygons is exponential.
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N points number of intersections = O( )
N
SPANNING PATH THEOREM
for any line
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N points number of intersections = O( )
N
SPANNING PATH THEOREM
for random line
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Join two closest points; remove; repeat
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number of intersections = O( )
N
for random line
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Difficulty 1: Produces a matching, not a simple polygon
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Matching Tree
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Remove each edge and one of its adjacent vertices
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number of intersections =
O( ) = O( ) 8/4/2/ NNNN
for random line
N
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Tree Hamiltonian Circuit
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(via DFS)
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Hamiltonian Circuit Simple Polygon
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(via edge switching)
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Simple Polygon
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for random line
number of intersections = O( )
N
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Change definition of randomness
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New definition: A random line joins 2 of the N points picked at random.
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NNext goal A random line cuts O( ) edges
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Euclidean is wrong metric
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prob [ line(random pair) cuts ab ]
b
a
New metric: d(a,b)=
pick random pair
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New metric has “dimension” 2
b
a
pick random pair
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NThis ensures that a random line cuts O( ) edges
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N
Final goal: A line between any 2 points picked
at cuts O( ) edges
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Increase d(a,b) multiplicatively (as in BOOSTING )
a
b
double probability of picking pair
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ANY line cuts O( ) edgesN
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Spanning Path Theorem
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APPLICATION: SIMPLEX RANGE COUNTING [2, p.214]
How many points in the triangle?
66
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Ray shooting in O(log N) time
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Ray shooting in O(log N) time
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APPLICATION: SIMPLEX RANGE COUNTING
How many points in the triangle?
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APPLICATION: SIMPLEX RANGE COUNTING
Triangle range counting in O( ) timeN~
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Spanning Path Theorem
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-APPROXIMATION (for triangles)
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Subset A such that:
any triangle T
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Subset A such that:
any triangle T
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Subset A such that:
any triangle T
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Size of A is O( ) Size of A is O( )
Independent of NIndependent of N
Better than random! Better than random!
Size of A is O( ) Size of A is O( )
Independent of NIndependent of N
Better than random! Better than random!
-4/3-4/3~
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Keep every other edge
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Keep every other edge
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Color randomly red/blue
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discrepancy within any triangle = ?
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discrepancy within any triangle = 1
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discrepancy within any triangle =
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discrepancy within any triangle =
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Remove red points
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Recolor
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Remove red points
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Repeat until O( ) points left~ -4/3
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Subset A such that:
any triangle T
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A is called an -approximationA is called an -approximation (for triangles)(for triangles)
A is called an -approximationA is called an -approximation (for triangles)(for triangles)
Its size O( ) is Its size O( ) is independent of N independent of N Its size O( ) is Its size O( ) is independent of N independent of N
-4/3-4/3
~
A is computable in poly(N)A is computable in poly(N) A is computable in poly(N)A is computable in poly(N)
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Set System (X,Set System (X, ) ) Set System (X,Set System (X, ) )
22 22 XXXX
VC dim = max |shattered set|VC dim = max |shattered set|VC dim = max |shattered set|VC dim = max |shattered set|
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VC dim = VC dim = 33 VC dim = VC dim = 33
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Unbounded VC dimension
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Bounded VC dim implies Bounded VC dim implies thatthat Bounded VC dim implies Bounded VC dim implies thatthat
Given any Y X, number Given any Y X, number ofofdistinct sets Y S, where distinct sets Y S, where S , is O(|Y| )S , is O(|Y| )
Given any Y X, number Given any Y X, number ofofdistinct sets Y S, where distinct sets Y S, where S , is O(|Y| )S , is O(|Y| )
cccc
Dual set system Dual set system dual shatter dual shatter exponentexponentDual set system Dual set system dual shatter dual shatter exponentexponent
primal shatter exponent
primal shatter exponent
easy to determine
easy to determine
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VC dim = VC dim = ?? VC dim = VC dim = ??
(points, ellipsoids) in d-(points, ellipsoids) in d-dimdim
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dual shatter function = O(N )dual shatter function = O(N )
(points, ellipsoids) in d-dim(points, ellipsoids) in d-dim
dd
by Thom-Milnorby Thom-Milnor
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Set System (V,Set System (V, S)S) Set System (V,Set System (V, S)S)
O( ) O( ) O( ) O( ) -2+2/(d+1)-2+2/(d+1)~
d= VC dimensiond= VC dimensiond= VC dimensiond= VC dimension
Size of -approximation isSize of -approximation isSize of -approximation isSize of -approximation is
or primal/dual shatter exponentor primal/dual shatter exponentor primal/dual shatter exponentor primal/dual shatter exponent
[2, p.179][2, p.179][2, p.179][2, p.179]
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Set System (V,Set System (V, S)S) Set System (V,Set System (V, S)S)
O( ) O( ) O( ) O( ) -2-2~
Size of -approximation isSize of -approximation isSize of -approximation isSize of -approximation is
Computable in O(N) Computable in O(N) poly( )poly( )Computable in O(N) Computable in O(N) poly( )poly( )
/1
In comp geom, random bits help In comp geom, random bits help with simplicity but not with with simplicity but not with complexitycomplexity
In comp geom, random bits help In comp geom, random bits help with simplicity but not with with simplicity but not with complexitycomplexity
[2, p.175][2, p.175][2, p.175][2, p.175]
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-cutting-cutting -cutting-cutting
N lines
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[2, p.204][2, p.204][2, p.204][2, p.204]
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ApplicationApplication Hopcroft’s problem Hopcroft’s problem [2, p.213][2, p.213]
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DualizeDualize point (a,b)point (a,b) line line aX+bY=1aX+bY=1
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RecurseRecurse
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Hopcroft’s problemHopcroft’s problem
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N lines
Standard samplingStandard sampling Standard samplingStandard sampling
How many lines?How many lines?
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Set System (X,Set System (X, ) ) Set System (X,Set System (X, ) )
XX = set of N lines= set of N lines XX = set of N lines= set of N lines
= = = =
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N lines
easy to do with an -easy to do with an -approximationapproximationeasy to do with an -easy to do with an -approximationapproximation
How many lines?How many lines?
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N lines
Product samplingProduct sampling Product samplingProduct sampling
How many vertices?How many vertices?
[2, p.183][2, p.183][2, p.183][2, p.183]
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N lines
Unbounded VC-dim: Unbounded VC-dim: yet can be yet can be done!done! Unbounded VC-dim: Unbounded VC-dim: yet can be yet can be done!done!
How many vertices?How many vertices?
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Convex hull of N points in RConvex hull of N points in RConvex hull of N points in RConvex hull of N points in Rdddd
[2, p.283]
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Voronoi diagram of N points in EVoronoi diagram of N points in EVoronoi diagram of N points in EVoronoi diagram of N points in Edddd
http://www.math.psu.edu/qdu/Res/Pic/gulf.jpg
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Linear programming in linear time Linear programming in linear time with fixed number of variableswith fixed number of variablesLinear programming in linear time Linear programming in linear time with fixed number of variableswith fixed number of variables
LP-type programming in linear LP-type programming in linear time with fixed number of time with fixed number of variablesvariables
[1, p.82]
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Linear programming in linear time Linear programming in linear time with fixed number of variableswith fixed number of variables
LP-type programming in linear LP-type programming in linear time with fixed number of time with fixed number of variablesvariables
LP-type programming in linear LP-type programming in linear time with fixed number of time with fixed number of variablesvariables [2, p.307]
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dSmallest ellipsoid enclosing N points in R
[2, p.313]
in O (N) time!d
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[0]
Sampling tool for approximate
geometric optimization
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2 Analytical tools2 Analytical tools 2 Analytical tools2 Analytical tools
2.1 randomized scaling2.1 randomized scaling
2.2 backward analysis2.2 backward analysis
2.1 randomized scaling2.1 randomized scaling
2.2 backward analysis2.2 backward analysis
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2 Analytical tools2 Analytical tools 2 Analytical tools2 Analytical tools
2.1 randomized scaling2.1 randomized scaling 2.1.2 k-sets2.1.2 k-sets 2.1.2 crossing lemma2.1.2 crossing lemma
2.2 backward analysis2.2 backward analysis
2.1 randomized scaling2.1 randomized scaling 2.1.2 k-sets2.1.2 k-sets 2.1.2 crossing lemma2.1.2 crossing lemma
2.2 backward analysis2.2 backward analysis
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2 Analytical tools2 Analytical tools 2 Analytical tools2 Analytical tools
2.1 randomized scaling2.1 randomized scaling 2.1.2 k-sets 2.1.2 k-sets 2.1.2 crossing lemma2.1.2 crossing lemma
2.2 backward analysis2.2 backward analysis
2.1 randomized scaling2.1 randomized scaling 2.1.2 k-sets 2.1.2 k-sets 2.1.2 crossing lemma2.1.2 crossing lemma
2.2 backward analysis2.2 backward analysis
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K-SETS
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n(i,j) = 9n(i,j) = 9
pp
i
j
f = { (i,j) | i<j and n(i,j)= f = { (i,j) | i<j and n(i,j)= k }k }kk
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f = 6f = 600
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TheoremTheorem: : TheoremTheorem: :
[4, p.141]
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X = 3X = 3
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TheoremTheorem: : TheoremTheorem: :
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TheoremTheorem: : TheoremTheorem: :
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2 Analytical tools2 Analytical tools 2 Analytical tools2 Analytical tools
2.1 randomized scaling2.1 randomized scaling 2.1.2 k-sets2.1.2 k-sets 2.1.2 crossing lemma 2.1.2 crossing lemma
2.2 backward analysis2.2 backward analysis
2.1 randomized scaling2.1 randomized scaling 2.1.2 k-sets2.1.2 k-sets 2.1.2 crossing lemma 2.1.2 crossing lemma
2.2 backward analysis2.2 backward analysis
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The Crossing Lemma: The Crossing Lemma: The Crossing Lemma: The Crossing Lemma:
[4, p.55]
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Pick each vertex with prob p
Set p= 4n/m
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Corollary:Corollary:Corollary:Corollary:# point/line incidences = O(N # point/line incidences = O(N ) ) # point/line incidences = O(N # point/line incidences = O(N ) )
4/34/34/34/3
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Corollary:Corollary:Corollary:Corollary:# unit-distance pairs = # unit-distance pairs = O(N ) O(N ) # unit-distance pairs = # unit-distance pairs = O(N ) O(N )
4/34/34/34/3
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2 Analytical tools2 Analytical tools 2 Analytical tools2 Analytical tools
2.1 2.1 randomized scalingrandomized scaling 2.1.2 k-sets2.1.2 k-sets 2.1.2 crossing lemma2.1.2 crossing lemma
2.2 backward analysis2.2 backward analysis
2.1 2.1 randomized scalingrandomized scaling 2.1.2 k-sets2.1.2 k-sets 2.1.2 crossing lemma2.1.2 crossing lemma
2.2 backward analysis2.2 backward analysis
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Linear ProgrammingLinear Programming Linear ProgrammingLinear Programming[1, p76]
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N constraints and d variablesN constraints and d variables
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N constraints and d variablesN constraints and d variables
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Planar GraphPlanar Graph
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Planar Separator TheoremPlanar Separator Theorem
Remove O( ) vertices Remove O( ) vertices (1/3-2/3) (1/3-2/3) cutcut
N
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[5, p96][5, p96]
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Stereographic liftingStereographic lifting
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Centerpoint Theorem
(1/3,2/3) cut
(1/4,3/4) cut in 3D
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Can assume centerpoint is center of sphere
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Can assume centerpoint is center of sphere
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BIBLIOGRAPHYBIBLIOGRAPHYBIBLIOGRAPHYBIBLIOGRAPHY
The results mentioned in this tutorial, as well as the history behind The results mentioned in this tutorial, as well as the history behind them, are discussed in detail in the surveys and monographs below.them, are discussed in detail in the surveys and monographs below.The results mentioned in this tutorial, as well as the history behind The results mentioned in this tutorial, as well as the history behind them, are discussed in detail in the surveys and monographs below.them, are discussed in detail in the surveys and monographs below.