2. Tools from Computational Geometry Bernard Chazelle Princeton University Bernard Chazelle Princeton University Tutorial FOCS 2005

3. Tools from Computational Geometry Bernard Chazelle Princeton University Bernard Chazelle Princeton University Tutorial FOCS 1905

5. Ruler & Compass Algorithms

6. Gauss: 17-gon

8. Constructing Regular N-gons 3folklore 5antiquity 17 Gauss (1796) 257 Richelot (1832) 65537 Hermes (1879) Gauss Fermat primes 2 2 k can’t do heptagons +1 proof covers a gym Hilbert proved lower bounds on number of steps

9. Tools from Computational Geometry Bernard Chazelle Princeton University Bernard Chazelle Princeton University Tutorial FOCS 2005

10. algorithmic algorithmic analytical analytical TOOLS FROM COMPUTATIONAL GEOMETRY TOOLS FROM COMPUTATIONAL GEOMETRY

11. 1 Algorithmic tools 1 Algorithmic tools geometric divide & conquer geometric divide & conquer

12. Voronoi Diagram

18. Works well also for convex hulls, nearest neighbors [3,6]

19. Works not so well for multidimensional searching: quadtrees, kd-trees: highly sub-optimal

20. Hopcroft’s problem Any point/line incidence?N points and N lines

21. Naïve divide & conquer

22. O(N logN) time Point location in line arrangement

23. ~ O(N ) time 3/2

24. 1 Algorithmic tools 1 Algorithmic tools geometric divide & conquer geometric divide & conquer

25. 1 Algorithmic tools 1 Algorithmic tools geometric divide & conquer geometric divide & conquer

26. [2, p.123]

27. N points

28. number of intersections = O( ) for any line

29. Often, the number of simple polygons is exponential.

30. Sometimes, it’s unique… Often, the number of simple polygons is exponential.

31. Sometimes, it’s unique… Often, the number of simple polygons is exponential.

32. N points number of intersections = O( ) SPANNING PATH THEOREM for any line

34. N points number of intersections = O( ) SPANNING PATH THEOREM for random line

35. Join two closest points; remove; repeat

37. number of intersections = O( ) for random line

38. Difficulty 1: Produces a matching, not a simple polygon

39. Matching  Tree

40. Remove each edge and one of its adjacent vertices

49. number of intersections = O( ) = O( ) for random line

50. Tree  Hamiltonian Circuit

51. (via DFS)

52. Hamiltonian Circuit  Simple Polygon

53. (via edge switching)

54. Simple Polygon

55. for random line number of intersections = O( )

56. Change definition of randomness

57. New definition: A random line joins 2 of the N points picked at random.

58. Next goal A random line cuts O( ) edges

59. Euclidean is wrong metric

60. prob [ line(random pair) cuts ab ] b a New metric: d(a,b)= pick random pair

61. New metric has “dimension” 2 b a pick random pair

62. This ensures that a random line cuts O( ) edges

63. Final goal: A line between any 2 points picked at cuts O( ) edges

64. Increase d(a,b) multiplicatively (as in BOOSTING ) a b double probability of picking pair

65. ANY line cuts O( ) edges

66. Spanning Path Theorem

67. APPLICATION: SIMPLEX RANGE COUNTING [2, p.214] How many points in the triangle? 6 6

68. Ray shooting in O(log N) time

70. APPLICATION: SIMPLEX RANGE COUNTING How many points in the triangle?

71. APPLICATION: SIMPLEX RANGE COUNTING Triangle range counting in O( ) time ~

72. Spanning Path Theorem

73. -APPROXIMATION (for triangles)

74. Subset A such that: any triangle T

75. Subset A such that: any triangle T

76. Subset A such that: any triangle T

77. Size of A is O( ) Independent of N Better than random! Size of A is O( ) Independent of N Better than random! -4/3 ~

79. Keep every other edge

81. Color randomly red/blue

82. discrepancy within any triangle = ?

83. discrepancy within any triangle = 1

84. discrepancy within any triangle =

86. Remove red points

87. Recolor

88. Remove red points

89. Repeat until O( ) points left ~ -4/3

92. Subset A such that: any triangle T

93. A is called an -approximation A is called an -approximation (for triangles) (for triangles) A is called an -approximation A is called an -approximation (for triangles) (for triangles) Its size O( ) is independent of N Its size O( ) is independent of N -4/3 ~ A is computable in poly(N) A is computable in poly(N)

95. Set System (X, ) Set System (X, ) 2 2 XX VC dim = max |shattered set|

96. VC dim = 3 VC dim = 3

97. Unbounded VC dimension

98. Bounded VC dim implies that Bounded VC dim implies that Given any Y X, number of distinct sets Y S, where S, is O(|Y| ) Given any Y X, number of distinct sets Y S, where S, is O(|Y| ) cc Dual set system  dual shatter exponent primal shatter exponent easy to determine

99. VC dim = ? VC dim = ? (points, ellipsoids) in d-dim (points, ellipsoids) in d-dim

100. dual shatter function = O(N ) dual shatter function = O(N ) (points, ellipsoids) in d-dim d by Thom-Milnor

101. Set System (V, S) Set System (V, S) O( ) -2+2/(d+1) ~ d= VC dimension Size of -approximation is or primal/dual shatter exponent [2, p.179]

102. Set System (V, S) Set System (V, S) O( ) -2 ~ Size of -approximation is Computable in O(N) poly( ) In comp geom, random bits help with simplicity but not with complexity [2, p.175]

104. -cutting -cutting N lines

105. [2, p.204]

106. Application Hopcroft’s problem [2, p.213]

108. Dualize point (a,b) line aX+bY=1

109. Recurse

110. Hopcroft’s problem

112. N lines Standard sampling Standard sampling How many lines?

113. Set System (X, ) Set System (X, ) X = set of N lines X = set of N lines = =

114. N lines easy to do with an -approximation How many lines?

115. N lines Product sampling Product sampling How many vertices? [2, p.183]

116. N lines Unbounded VC-dim: yet can be done! Unbounded VC-dim: yet can be done! How many vertices?

118. Convex hull of N points in R dd [ 2, p.283]

119. Voronoi diagram of N points in E dd http://www.math.psu.edu/qdu/Res/Pic/gulf.jpg

120. Linear programming in linear time with fixed number of variables LP-type programming in linear time with fixed number of variables [1, p.82]

121. Linear programming in linear time with fixed number of variables LP-type programming in linear time with fixed number of variables [2, p.307]

122. d Smallest ellipsoid enclosing N points in R [2, p.313] in O (N) time! d

123. [0] Sampling tool for approximate geometric optimization

124. 2 Analytical tools 2 Analytical tools 2.1 randomized scaling 2.2 backward analysis 2.1 randomized scaling 2.2 backward analysis

125. 2 Analytical tools 2 Analytical tools 2.1 randomized scaling 2.1.2 k-sets 2.1.2 crossing lemma 2.2 backward analysis 2.1 randomized scaling 2.1.2 k-sets 2.1.2 crossing lemma 2.2 backward analysis

127. 2 Analytical tools 2 Analytical tools 2.1 randomized scaling 2.1.2 k-sets 2.1.2 crossing lemma 2.2 backward analysis 2.1 randomized scaling 2.1.2 k-sets 2.1.2 crossing lemma 2.2 backward analysis

128. K-SETS

129. n(i,j) = 9 p p i j f = { (i,j) | i<j and n(i,j)= k } k

130. f = 6 0

131. Theorem:Theorem: [4, p.141]

132. X = 3

133. Theorem:Theorem:

134. Theorem:Theorem:

135. 2 Analytical tools 2 Analytical tools 2.1 randomized scaling 2.1.2 k-sets 2.1.2 crossing lemma 2.2 backward analysis 2.1 randomized scaling 2.1.2 k-sets 2.1.2 crossing lemma 2.2 backward analysis

136. The Crossing Lemma: [4, p.55]

137. Pick each vertex with prob p Set p= 4n/m

138. Corollary:Corollary: # point/line incidences = O(N ) 4/34/3

139. Corollary:Corollary: # unit-distance pairs = O(N ) 4/34/3

140. 2 Analytical tools 2 Analytical tools 2.1 randomized scaling 2.1.2 k-sets 2.1.2 crossing lemma 2.2 backward analysis 2.1 randomized scaling 2.1.2 k-sets 2.1.2 crossing lemma 2.2 backward analysis

141. Linear Programming Linear Programming [1, p76]

150. N constraints and d variables

153. Planar Graph

154. Planar Separator Theorem Remove O( ) vertices  (1/3-2/3) cut

155. [5, p96]

159. Stereographic lifting

162. Centerpoint Theorem (1/3,2/3) cut (1/4,3/4) cut in 3D

163. Can assume centerpoint is center of sphere

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