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20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
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20110306 csseminar cg_illustrations_vyatkina
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20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
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20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
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20110306 csseminar cg_illustrations_vyatkina
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20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
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20110306 csseminar cg_illustrations_vyatkina
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20110306 csseminar cg_illustrations_vyatkina
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20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
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20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
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20110306 csseminar cg_illustrations_vyatkina
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20110306 csseminar cg_illustrations_vyatkina
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20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
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20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
20110306 csseminar cg_illustrations_vyatkina
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20110306 csseminar cg_illustrations_vyatkina

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  • 1. þþ þ ¹ ¾¼½½ º þ þ
  • 2. þ½ þ¾¿ þ þ
  • 3. þS ⊂ R2 þ þ
  • 4. þ S ⊂ R2þ CH(S) S ¸ S þ þ
  • 5. þ S ⊂ R2þ CH(S) S ¸ S CH(S) ¸ S þ þ
  • 6. þP n þ þ
  • 7. þ P nCH(P) S þ þ
  • 8. þ ü þ þ
  • 9. þ ü þ þ
  • 10. þ ü þ þ
  • 11. þ ü Ω(n log n) þ þ
  • 12. þ
  • 13. þ þ
  • 14. þ
  • 15. p1 þ þ
  • 16. þ
  • 17. p1 þ þ
  • 18. þ
  • 19. p8 p5 p6 p7 p4 p3 p9 p2 p1 þ þ
  • 20. þ
  • 21. p8 p5 p6 p7 p4 p3 p9 p2 p1 þ þ
  • 22. þ
  • 23. p8 p5 p6 p7 p4 p3 p9 p2 p1 þ þ
  • 24. þ
  • 25. p8 p5 p6 p7 p4 p3 p9 p2 p1 þ þ
  • 26. þ
  • 27. p8 p5 p6 p7 p4 p3 p9 p2 p1 þ þ
  • 28. þ
  • 29. p8 p5 p6 p7 p4 p3 p9 p2 p1 þ þ
  • 30. þ
  • 31. p8 p5 p6 p7 p4 p3 p9 p2 p1 þ þ
  • 32. þ
  • 33. p8 p5 p6 p7 p4 p3 p9 p2 p1 þ þ
  • 34. þ
  • 35. p8 p5 p6 p7 p4 p3 p9 p2 p1 þ þ
  • 36. þ
  • 37. p8 p5 p6 p7 p4 p3 p9 p2 p1 þ þ
  • 38. þ
  • 39. p8 p5 p6 p7 p4 p3 p9 p2 p1 þ þ
  • 40. þ
  • 41. p8 p5 p6 p7 p4 p3 p9 p2 p1 þ þ
  • 42. þ
  • 43. p8 p5 p6 p7 p4 p3 p9 p2 p1 þ þ
  • 44. þ
  • 45. p8 p5 p6 p7 p4 p3 p9 p2 p1 þ þ
  • 46. þ
  • 47. p8 p5 p6 p7 p4 p3 p9 p2 p1 þ þ
  • 48. þ
  • 49. p8 p5 p6 p7 p4 p3 p9 p2 p1 þ þ
  • 50. þ
  • 51. p8 p5 p6 p7 p4 p3 p9 p2 p1 þ þ
  • 52. þ
  • 53. p8 p5 p6 p7 p4 p3 p9 p2 p1 þ þ
  • 54. þ
  • 55. p8 p5 p6 p7 p4 p3 p9 p2 p1 þ þ
  • 56. þ
  • 57. p8 p5 p6 p7 p4 p3 p9 p2 p1 þ þ
  • 58. þ
  • 59. p8 p5 p6 p7 p4 p3 p9 p2 p1 þ þ
  • 60. þ
  • 61. þ þ þ
  • 62. þ
  • 63. þ þ þ
  • 64. þ
  • 65. þ Pi = {p1 , . . . , pi } þ þ
  • 66. þ
  • 67. þ Pi = {p1 , . . . , pi } i¹ CH(Pi ) þ þ
  • 68. þ
  • 69. þ Pi = {p1 , . . . , pi } i¹ CH(Pi )þ ¡ O(n log n) þ þ
  • 70. þ
  • 71. þ Pi = {p1 , . . . , pi } i¹ CH(Pi )þ ¡ O(n log n) p1 O(n) þ þ
  • 72. þ
  • 73. þ Pi = {p1 , . . . , pi } i¹ CH(Pi )þ ¡ O(n log n) p1 O(n) O(n log n) þ þ
  • 74. þ
  • 75. þ Pi = {p1 , . . . , pi } i¹ CH(Pi )þ ¡ O(n log n) p1 O(n) O(n log n) O(n) þ þ
  • 76. þ
  • 77. þ Pi = {p1 , . . . , pi } i¹ CH(Pi )þ ¡ O(n log n) p1 O(n) O(n log n) O(n) þ þ þ
  • 78. þ
  • 79. þ Pi = {p1 , . . . , pi } i¹ CH(Pi )þ ¡ O(n log n) p1 O(n) O(n log n) O(n) þ p2 ¸ ººº¸ pn þ þ
  • 80. þ
  • 81. þ Pi = {p1 , . . . , pi } i¹ CH(Pi )þ ¡ O(n log n) p1 O(n) O(n log n) O(n) þ p2 ¸ ººº¸ pn ¸ n−1 þ þ
  • 82. þ
  • 83. þ Pi = {p1 , . . . , pi } i¹ CH(Pi )þ ¡ O(n log n) p1 O(n) O(n log n) O(n) þ p2 ¸ ººº¸ pn ¸ n−1 O(n) þ þ
  • 84. þ þ þ
  • 85. þ p1 þ þ
  • 86. þ p1 þ þ
  • 87. þ p1 þ þ
  • 88. þ p2 p1 þ þ
  • 89. þ p2 p1 þ þ
  • 90. þ p2 p1 þ þ
  • 91. þ p3 p2 p1 þ þ
  • 92. þ p3 p2 p1 þ þ
  • 93. þ p4 p3 p2 p1 þ þ
  • 94. þ p4 p5 p3 p2 p1 þ þ
  • 95. þ p4 p5 p6 p3 p2 p1 þ þ
  • 96. þ p4 p5 p6 p3 p2 p1 þ þ
  • 97. þ þ þ
  • 98. þ CH(P) þ þ
  • 99. þ CH(P)þ ¡ O(nh)¸ h CH(P) þ þ
  • 100. þ CH(P)þ ¡ O(nh)¸ h CH(P) ü ¸ þ þ
  • 101. þ CH(P)þ ¡ O(nh)¸ h CH(P) ü ¸ O(n) þ þ
  • 102. þ½ ¾
  • 103. ¸ O(n log n) þ þ
  • 104. þ½ ¾
  • 105. ¸ O(n log n)½ ¿ ¸ O(nh) þ þ
  • 106. þ½ ¾
  • 107. ¸ O(n log n)½ ¿ ¸ O(nh)½ ¸ O(n log h) þ þ
  • 108. þ½ ¾
  • 109. ¸ O(n log n)½ ¿ ¸ O(nh)½ ¸ O(n log h) þ þ
  • 110. þ½ ¾
  • 111. ¸ O(n log n)½ ¿ ¸ O(nh)½ ¸ O(n log h)½ ¸ O(n log h) þ þ
  • 112. þü m≤n þ þ
  • 113. þü m≤n n r = ⌈m⌉ þ þ
  • 114. þü m≤n n r = ⌈m⌉ P r P1 ¸ º º º ¸ Pr |P1 | = · · · = |Pr −1 | = m¸ |Pr | ≤ m þ þ
  • 115. þü m≤n n r = ⌈m⌉ P r P1 ¸ º º º ¸ Pr |P1 | = · · · = |Pr −1 | = m¸ |Pr | ≤ m Pi CH(Pi )
  • 116. þ þ
  • 117. þü m≤n n r = ⌈m⌉ P r P1 ¸ º º º ¸ Pr |P1 | = · · · = |Pr −1 | = m¸ |Pr | ≤ m Pi CH(Pi )
  • 118. þ þ
  • 119. þü þ þ
  • 120. þü þ þ
  • 121. þü þ þ
  • 122. þü p1 þ þ
  • 123. þü p1 þ þ
  • 124. þü p1 þ þ
  • 125. þü p2 p1 þ þ
  • 126. þü p2 p1 þ þ
  • 127. þ þ þ
  • 128. þ CH(Pi ) O(m log m) þ þ
  • 129. þ CH(Pi ) O(m log m)O(rm log m) = O(n log m) þ þ
  • 130. þ CH(Pi ) O(m log m)O(rm log m) = O(n log m) þ þ
  • 131. þ CH(Pi ) O(m log m)O(rm log m) = O(n log m) CH(Pi ) O(log m) þ þ
  • 132. þ CH(Pi ) O(m log m)O(rm log m) = O(n log m) CH(Pi ) O(log m) CH(P) O(r log m) þ þ
  • 133. þ CH(Pi ) O(m log m)O(rm log m) = O(n log m) CH(Pi ) O(log m) CH(P) O(r log m) CH(P) nO(hr log m) = O(h · m · log m) þ þ
  • 134. þ CH(Pi ) O(m log m)O(rm log m) = O(n log m) CH(Pi ) O(log m) CH(P) O(r log m) CH(P) nO(hr log m) = O(h · m · log m) n O n+h· m log m þ þ
  • 135. þ CH(Pi ) O(m log m)O(rm log m) = O(n log m) CH(Pi ) O(log m) CH(P) O(r log m) CH(P) nO(hr log m) = O(h · m · log m) n O n+h· m log mm = h O(n log h) þ þ
  • 136. þÈ ÖØ ÐÀÙÐдP ¸mµ½ n r ← ⌈m⌉¾ P → P1 , . . . , Pr¿ ÓÖ i ← 1 ØÓ r Ó CH(Pi ) p1 ← P ´ µ ÓÖ k ← 1 ØÓ m Ó pk+1 CH(P) pk+1 = p1 Ø Ò Ö ØÙÖÒ ´ ´p1 ¸ º º º ¸ pk µ µ½¼ Ö ØÙÖÒ m þ þ
  • 137. þÈ ÖØ ÐÀÙÐдP ¸mµ½ n r ← ⌈m⌉¾ P → P1 , . . . , Pr¿ ÓÖ i ← 1 ØÓ r £ O(r · m log m) = O(n log m) Ó CH(Pi ) p1 ← P ´ µ ÓÖ k ← 1 ØÓ m £ O(m · r log m) = O(n log m) Ó pk+1 CH(P) pk+1 = p1 Ø Ò Ö ØÙÖÒ ´ ´p1 ¸ º º º ¸ pk µ µ½¼ Ö ØÙÖÒ m þ þ
  • 138. þÈ ÖØ ÐÀÙÐдP ¸mµ½ n r ← ⌈m⌉¾ P → P1 , . . . , Pr¿ ÓÖ i ← 1 ØÓ r Ó CH(Pi ) p1 ← P ´ µ ÓÖ k ← 1 ØÓ m Ó pk+1 CH(P) pk+1 = p1 Ø Ò Ö ØÙÖÒ ´ ´p1 ¸ º º º ¸ pk µ µ½¼ Ö ØÙÖÒ mþ O(n log m) þ þ
  • 139. þ ÓÒÚ ÜÀÙÐдP µ½ ÓÖ t ← 1, 2, 3, . . . Ó m ← min{22 , n} t¾¿ L ← È ÖØ ÐÀÙÐдP ¸mµ L= m Ø Ò Ö ØÙÖÒ L þ þ
  • 140. þ ÓÒÚ ÜÀÙÐдP µ½ ÓÖ t ← 1, 2, 3, . . . Ó m ← min{22 , n} t¾¿ L ← È ÖØ ÐÀÙÐдP ¸mµ L= m Ø Ò Ö ØÙÖÒ Lü t = ⌈log log h⌉ þ þ
  • 141. þü t þ t O(n log 22 ) = O(n · 2t ) þ þ
  • 142. þü t þ t O(n log 22 ) = O(n · 2t ) ⌈log log h⌉ O(n · 2t ) t=1 þ þ
  • 143. þü t þ t O(n log 22 ) = O(n · 2t ) ⌈log log h⌉ O(n · 2t ) t=1 ⌈log log h⌉ n · 2t n · 2⌈log log h⌉+1 4n · 2log log h = 4n log h t=1 þ þ
  • 144. þü t þ t O(n log 22 ) = O(n · 2t ) ⌈log log h⌉ O(n · 2t ) t=1 ⌈log log h⌉ n · 2t n · 2⌈log log h⌉+1 4n · 2log log h = 4n log h t=1 þ ¡ O(n log h) þ þ
  • 145. þü t þ t O(n log 22 ) = O(n · 2t ) ⌈log log h⌉ O(n · 2t ) t=1 ⌈log log h⌉ n · 2t n · 2⌈log log h⌉+1 4n · 2log log h = 4n log h t=1 þ ¡ O(n log h) O(n) þ þ
  • 146. þ½ þ¾¿ þ þ
  • 147. þ P n þ þ
  • 148. þ P n ¸ P þ þ
  • 149. þ P þ þ
  • 150. þ P P¸ ¸ þ þ
  • 151. þ P P¸ ¸ P þ þ
  • 152. þ1≤i ≤n þ þ
  • 153. þ 1≤i ≤nPi = {p1 , . . . , pi } þ þ
  • 154. þ 1≤i ≤nPi = {p1 , . . . , pi }Ci Pi þ þ
  • 155. þ
  • 156. Ê Ò ÓÑÈ ÖÑÙØ Ø ÓÒ´Aµ ½ ÓÖ k ← n ÓÛÒØÓ ¾ ¾ Ó rndIndex ← Ê Ò ÓÑ´k µ ¿ A[k] ↔ A[rndIndex] A ¸ Ê Ò ÓÑ´k µ [1, k] O(1) þ þ
  • 157. þ
  • 158. Ê Ò ÓÑÈ ÖÑÙØ Ø ÓÒ´Aµ ½ ÓÖ k ← n ÓÛÒØÓ ¾ ¾ Ó rndIndex ← Ê Ò ÓÑ´k µ ¿ A[k] ↔ A[rndIndex] A ¸ Ê Ò ÓÑ´k µ [1, k] O(1) ∀a ∈ A¸ a j 1/n¸ 1≤j ≤n þ þ
  • 159. þ½ pi Ci −1 ¸ Ci = Ci −1 pi Ci −1 ¸ Ci pi þ þ
  • 160. þ½ pi Ci −1 ¸ Ci = Ci −1 pi Ci −1 ¸ Ci pi pi+1 Ci+1 pi Ci-1=Ci þ þ
  • 161. þÅÒ ÖÐ ´P µ½ P¾ C2 ← p1 p2¿ ÓÖ i ← 3 ØÓ n Ó pi Ci −1 Ø Ò Ci ← Ci −1 Ð× Ci ← Å Ò ÖÐ ¹½´Pi −1 ¸ pi µ þ þ
  • 162. þ ¾q ¸ ¸ P¸ q þ þ
  • 163. þ ¾q ¸ ¸ P¸ qCiq Pi ¸ q þ þ
  • 164. þ ¾q ¸ ¸ P¸ qCiq Pi ¸ q pi Ciq ¸ −1 Ciq = Ciq −1 pi Ciq ¸ −1 Ciq pi þ þ
  • 165. þ ÅÒ ÖÐ ¹½´P ¸q µ½ P¾ C1 ← p1 q¿ ÓÖ j ← 2 ØÓ n Ó pj Cj−1 Ø Ò Cj ← Cj−1 Ð× Cj ← Å Ò ÖÐ ¹¾´Pj−1 ¸ pj ¸ q µ þ þ
  • 166. þ ¿q1 ¸ q2 ¸ ¸ P¸ q1 q2 þ þ
  • 167. þ ¿q1 ¸ q2 ¸ ¸ P¸ q1 q2Ciq1 ,q2 Pi ¸ q1 q2 þ þ
  • 168. þ ¿q1 ¸ q2 ¸ ¸ P¸ q1 q2Ciq1 ,q2 Pi ¸ q1 q2 pi Ciq1 ,q2 ¸ −1 Ciq1 ,q2 = Ciq1 ,q2 −1 pi Ciq1 ,q2 ¸ −1 Ciq1 ,q2 pi þ þ
  • 169. þ ÅÒ ÖÐ ¹¾´P ¸q1 ¸ q2 µ½ C0 ← q1 q2¾ ÓÖ k ← 1 ØÓ n¿ Ó pk Ck−1 Ø Ò Ck ← Ck−1 Ð× Ck ← ¸ q1 ¸ q2 pk þ þ
  • 170. þü ÅÒ ÖÐ ¹¾´P µ þ þ
  • 171. þü ÅÒ ÖÐ ¹¾´P µ þ O(n) þ þ
  • 172. þü ÅÒ ÖÐ ¹¾´P µ þ O(n) ÅÒ ÖÐ ¹½´P µ þ þ
  • 173. þü ÅÒ ÖÐ ¹¾´P µ þ O(n) ÅÒ ÖÐ ¹½´P µ þ Å Ò ÖÐ ¹¾ j 2/j þ þ
  • 174. þü ÅÒ ÖÐ ¹¾´P µ þ O(n) ÅÒ ÖÐ ¹½´P µ þ Å Ò ÖÐ ¹¾ j 2/j n O(n) + O(j) 2 = O(n) j j=2 þ þ
  • 175. þü ÅÒ ÖÐ ¹¾´P µ þ O(n) ÅÒ ÖÐ ¹½´P µ þ Å Ò ÖÐ ¹¾ j 2/j n O(n) + O(j) 2 = O(n) j j=2 ÅÒ ÖÐ ´P µ þ þ
  • 176. þü ÅÒ ÖÐ ¹¾´P µ þ O(n) ÅÒ ÖÐ ¹½´P µ þ Å Ò ÖÐ ¹¾ j 2/j n O(n) + O(j) 2 = O(n) j j=2 ÅÒ ÖÐ ´P µ þ Å Ò ÖÐ ¹½ i 3/i þ þ
  • 177. þü ÅÒ ÖÐ ¹¾´P µ þ O(n) ÅÒ ÖÐ ¹½´P µ þ Å Ò ÖÐ ¹¾ j 2/j n O(n) + O(j) 2 = O(n) j j=2 ÅÒ ÖÐ ´P µ þ Å Ò ÖÐ ¹½ i 3/i n O(n) + O(i) 3 = O(n) i i =3 þ þ
  • 178. þü ÅÒ ÖÐ ¹¾´P µ þ O(n) ÅÒ ÖÐ ¹½´P µ þ Å Ò ÖÐ ¹¾ j 2/j n O(n) + O(j) 2 = O(n) j j=2 ÅÒ ÖÐ ´P µ þ Å Ò ÖÐ ¹½ i 3/i n O(n) + O(i) 3 = O(n) i i =3 O(n) þ þ
  • 179. þü ÅÒ ÖÐ ¹¾´P µ þ O(n) ÅÒ ÖÐ ¹½´P µ þ Å Ò ÖÐ ¹¾ j 2/j n O(n) + O(j) 2 = O(n) j j=2 ÅÒ ÖÐ ´P µ þ Å Ò ÖÐ ¹½ i 3/i n O(n) + O(i) 3 = O(n) i i =3 O(n) O(n) þ þ
  • 180. þ½ þ¾¿ þ þ
  • 181. þ P n þ þ
  • 182. þ P n P þ þ
  • 183. þ þ þ
  • 184. þ þ þ
  • 185. þ p = (a, b) → ℓ y = ax + bℓ y = kx + d → p = (−k, d) y y x x þ þ
  • 186. þ p = (a, b) → ℓ y = ax + bℓ y = kx + d → p = (−k, d) y y p1 x x l1 þ þ
  • 187. þ p = (a, b) → ℓ y = ax + bℓ y = kx + d → p = (−k, d) y y p2 p1 x l2 x l1 þ þ
  • 188. þ p = (a, b) → ℓ y = ax + bℓ y = kx + d → p = (−k, d) y y p2 p1 x l2 x p3 l3 l1 þ þ
  • 189. þ p = (a, b) → ℓ y = ax + b ℓ y = kx + d → p = (−k, d) y y l4 p2 p1 x l2 xp4 p3 l3 l1 þ þ
  • 190. þ p = (a, b) → ℓ y = ax + b ℓ y = kx + d → p = (−k, d) y y l4 p2 p13 p1 x l2 xp4 p3 l13 l3 l1 þ þ
  • 191. þ p = (a, b) → ℓ y = ax + bℓ y = kx + d → p = (−k, d) y yp5 p13 p1 x x l5 p3 l13 l3 l1 þ þ
  • 192. þ p = (a, b) → ℓ y = ax + bℓ y = kx + d → p = (−k, d) y y hp5 p13 h p1 x x l5 p3 l13 l3 l1 þ þ
  • 193. þ þ þ
  • 194. þ△pi pj pk þ þ
  • 195. þ lk pij△pi pj pk pij ¸ lk þ þ
  • 196. þ lk pij△pi pj pk pij ¸ lk O(n) þ þ
  • 197. þ lk pij △pi pj pk pij ¸ lk O(n)þ ¡ O(n2 ) þ þ
  • 198. þ lk pij △pi pj pk pij ¸ lk O(n)þ ¡ O(n2 ) O(n2 ) þ þ
  • 199. þ ½ ź Ö ¸ Ǻ ÓÒ ¸ ź Ú Ò ÃÖ Ú Ð ¸ ź ÇÚ ÖÑ Ö׸ ÓÑÔÙØ Ø ÓÒ Ð ÓÑ ØÖÝ Ð ÓÖ Ø Ñ× Ò ÔÔÐ Ø ÓÒ× ¸ Ì Ö Ø ÓÒ¸ ËÔÖ Ò Ö¸ ¾¼¼ º¾ º dzÊÓÙÖ ¸ ÓÑÔÙØ Ø ÓÒ Ð ÓÑ ØÖÝ Ò ¸ Ë ÓÒ Ø ÓÒ¸ Ñ Ö ÍÒ Ú Ö× ØÝ ÈÖ ×׸ ½ º¿ º¹ º Ó ××ÓÒÒ Ø¸ ź Ú Ò ¸ ¡ÓÑ¡ØÖ Ð ÓÖ Ø Ñ ÕÙ ¸ × Ò ÒØ ÖÒ Ø ÓÒ Ð¸ È Ö ×¸ ½ º º ¸ º ¸ þ ¸ º º¸ º¸ ¸½ º º ÅÓÙÒظ ÓÑÔÙØ Ø ÓÒ Ð ÓÑ ØÖÝ Ä ØÙÖ ÆÓØ × ¸ ÐÐ ¾¼¼¾¸ ØØÔ »»ÛÛۺ׺ÙÑ º Ù» ÑÓÙÒØ» »Ä Ø×» Ð Ø×ºÔ þ þ
  • 200. þ½ ʺ ĺ Ö Ñ¸ Ò ÒØ Ð ÓÖ Ø Ñ× ÓÖ Ø ÖÑ Ò Ò Ø ÓÒÚ Ü ÙÐÐ Ó Ò Ø ÔÐ Ò Ö × Ø ¸ ÁÒ Óº ÈÖÓº Ä Øغ¸ ½ ½¿¾ ½¿¿¸ ½ ¾º¾ ʺ º  ÖÚ ×¸ ÇÒ Ø ÒØ Ø ÓÒ Ó Ø ÓÒÚ Ü ÙÐÐ Ó ÒØ × Ø Ó ÔÓ ÒØ× Ò Ø ÔÐ Ò ¸ ÁÒ Óº ÈÖÓº Ä Øغ¸ ¾ ½ ¾½¸ ½ ¿º¿ º º Ã Ö Ô ØÖ ¸ ʺ Ë Ð¸ Ì ÙÐØ Ñ Ø ÔÐ Ò Ö ÓÒÚ Ü ÙÐÐ Ð ÓÖ Ø Ñ ¸ ËÁ Šº ÓÑÔÙغ¸ ½ ´½µ ¾ ¾ ¸½ º ̺ Ò¸ ÇÔØ Ñ Ð ÓÙØÔÙØ¹× Ò× Ø Ú ÓÒÚ Ü ÙÐÐ Ð ÓÖ Ø Ñ× Ò ØÛÓ Ò Ø Ö Ñ Ò× ÓÒ× ¸ ×Öº ÓÑÔº ÓѺ¸ ½ ¿ ½ ¿ ¸ ½ º º Ï ÐÞи ËÑ ÐÐ ×Ø ÒÐÓ× Ò × × ´ ÐÐ× Ò ÐÐ Ô×Ó ×µ ¸ Ò Àº Å ÙÖ Ö ´ ºµ¸ Æ Û Ê ×ÙÐØ× Ò Æ Û ÌÖ Ò × Ò ÓÑÔÙØ Ö Ë Ò ¸ ÄÆ Ë¸ ¿ ¿ ¼¸ ËÔÖ Ò Ö¸ ½ ½º º Þ ÐÐ ¸ ĺ º ٠׸ º ̺ Ä ¸ Ì ÔÓÛ Ö Ó ÓÑ ØÖ Ù Ð ØÝ ¸ Á̸ ¾ ¼¸ ½ º þ þ

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