Tools from Computational Geometry

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Tools from Computational Geometry

• Bernard Chazelle
• Princeton University

Tutorial FOCS 2005
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Tools from Computational Geometry

• Bernard Chazelle
• Princeton University

Tutorial FOCS 1905
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Ruler Compass Algorithms
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Gauss 17-gon
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Gauss 17-gon
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Constructing Regular N-gons
k
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Gauss Fermat primes 2
1
3 folklore
5 antiquity
17 Gauss (1796)
257 Richelot (1832)
65537 Hermes (1879)
cant do heptagons
proof coversa gym
Hilbert proved lower bounds on number of steps
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Tools from Computational Geometry

• Bernard Chazelle
• Princeton University

Tutorial FOCS 2005
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algorithmic
TOOLS FROM COMPUTATIONAL GEOMETRY
analytical
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1 Algorithmic tools
geometric divide conquer
old style
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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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yes
Hopcrofts problem
Any point/line incidence?
N points and N lines
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Naïve divide conquer
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Point location in line arrangement
O(N logN) time
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3/2
O(N ) time
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1 Algorithmic tools
geometric divide conquer
old style
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1 Algorithmic tools
geometric divide conquer
new style
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Spanning path thm
2, p.123
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N points
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Optimal
for any line
number of intersections O( )
N points
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Often, the number of simple polygons is
exponential.
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Often, the number of simple polygons is
exponential.
Sometimes, its unique
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Often, the number of simple polygons is
exponential.
Sometimes, its unique
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SPANNING PATH THEOREM
for any line
number of intersections O( )
N points
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Proof
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SPANNING PATH THEOREM
for random line
number of intersections O( )
N points
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Join two closest points remove repeat
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for random line
number of intersections O( )
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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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for random line
number of intersections O(
) O( )
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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( )
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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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Next goal A random line cuts O( ) edges
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Euclidean is wrong metric
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b
pick random pair
a
prob line(random pair) cuts ab
New metric d(a,b)
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b
pick random pair
a
New metric has dimension 2
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This ensures that a random line cuts O( )
edges
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Final goal A line between any 2 points
picked at cuts O( ) edges
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b
a
double probability of picking pair
Increase d(a,b) multiplicatively (as in BOOSTING )
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QED
ANY line cuts O( ) edges
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Spanning Path Theorem
Application 1
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APPLICATION SIMPLEX RANGE COUNTING 2, p.214
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How many points in the triangle?
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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
QED

Triangle range counting in O( ) time
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Spanning Path Theorem
Application 2
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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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-4/3
Size of A is O( ) Independent of N
Better than random!
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Proof
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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
Optimal
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Remove red points
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Recolor
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Remove red points
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-4/3
Repeat until O( ) points left
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Subset A such that
any triangle T
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A is called an -approximation (for
triangles)

-4/3
Its size O( ) is independent of N

A is computable in poly(N)
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- approximation
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Set System (X, )
X
2
VC dim max shattered set
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VC dim 3
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Unbounded VC dimension
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Bounded VC dim implies that
Given any Y X, number of distinct sets Y
S, where S , is O(Y )
c
primal shatter exponent
easy to determine
Dual set system ? dual shatter exponent
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VC dim ?
(points, ellipsoids) in d-dim
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(points, ellipsoids) in d-dim
by Thom-Milnor
d
dual shatter function O(N )
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Set System (V, S)
d VC dimension
or primal/dual shatter exponent
Size of -approximation is

optimal
-22/(d1)
O( )
2, p.179
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Set System (V, S)
Size of -approximation is

-2
O( )
Computable in O(N) poly( )
Huge!
In comp geom, random bits help with simplicity
but not with complexity
2, p.175
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- cutting
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-cutting
N lines
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2, p.204
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Application Hopcrofts problem
2, p.213
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Dualize point (a,b)
line aXbY1
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Recurse
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Hopcrofts problem
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Product sampling
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Standard sampling
How many lines?
N lines
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Set System (X, )
X set of N lines

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easy to do with an -approximation
How many lines?
N lines
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Product sampling
2, p.183
How many vertices?
N lines
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Unbounded VC-dim yet can be done!
How many vertices?
N lines
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Sampling at work
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2, p.283
d
Convex hull of N points in R
Optimal
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d
Voronoi diagram of N points in E
http//www.math.psu.edu/qdu/Res/Pic/gulf.jpg
Optimal
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Linear programming in linear time with fixed
number of variables
1, p.82
LP-type programming in linear time with fixed
number of variables
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Linear programming in linear time with fixed
number of variables
LP-type programming in linear time with fixed
number of variables
2, p.307
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d
Smallest ellipsoid enclosing N points in R
in O (N) time!
d
2, p.313
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The CoreSet
Sampling tool for approximate geometric
optimization
0
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2 Analytical tools
2.1 randomized scaling
2.2 backward analysis
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2 Analytical tools
2.1 randomized scaling 2.1.2
k-sets 2.1.2 crossing lemma2.2
backward analysis
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2 Analytical tools
2.1 randomized scaling 2.1.2
k-sets 2.1.2 crossing lemma2.2
backward analysis
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K-SETS
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f (i,j) iltj and n(i,j) k
k
p
i
p
j
n(i,j) 9
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f 6
0
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4, p.141
Theorem
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X 3
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Theorem
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Theorem
QED
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2 Analytical tools
2.1 randomized scaling 2.1.2
k-sets 2.1.2 crossing lemma2.2
backward analysis
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The Crossing Lemma
4, p.55
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Pick each vertex with prob p
Set p 4n/m
QED
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4/3
Corollary
point/line incidences O(N )
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4/3
Corollary
unit-distance pairs O(N )
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2 Analytical tools
2.1 randomized scaling 2.1.2
k-sets 2.1.2 crossing lemma2.2
backward analysis
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Linear Programming
1, p76
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N constraints and d variables
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N constraints and d variables
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Geometrization



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Planar Graph
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Planar Separator Theorem
Remove O( ) vertices ? (1/3-2/3) cut
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A geometric proof



5, p96
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Stereographic lifting
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Centerpoint Theorem
(1/4,3/4) cut in 3D
(1/3,2/3) cut
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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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Thanks!



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BIBLIOGRAPHY
The results mentioned in this tutorial, as well
as the history behind them, are discussed in
detail in the surveys and monographs below.