Lecture 16

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Lecture 16

  1. 1. Randomized Algorithms CS648 Lecture 16 Randomized Incremental Construction (Backward analysis) 1
  2. 2. PROBLEM 1 FIND-MIN PROBLEM 2
  3. 3. Find-Min algorithm A 1 2 … ?? 3
  4. 4. A 1 2 … 4
  5. 5. 5
  6. 6. A 1 2 … 6
  7. 7. Fact: A is permuted randomly uniformly 7
  8. 8. A 1 2 … 8
  9. 9. PROBLEM 2 CLOSEST PAIR OF POINTS 9
  10. 10. Closest Pair of Points 10
  11. 11. 11
  12. 12. O(1) O(1) O(1) 12
  13. 13. 13
  14. 14. 14
  15. 15. 15
  16. 16. 16
  17. 17. 17
  18. 18. 18
  19. 19. RANDOMIZED INCREMENTAL CONSTRUCTION 19
  20. 20. Randomized Incremental Construction 20
  21. 21. Randomized Incremental Construction • Convex Hull of a set of points • Trapezoidal decomposition of a set of segments. • Convex polytope of a set of half-planes • Smallest sphere enclosing a set of points. • Linear programming in finite dimensions. 21
  22. 22. PROBLEM 3 CONVEX HULL OF POINTS 22
  23. 23. Convex hull of Points 23
  24. 24. Convex hull of Points 24
  25. 25. Randomized algorithm for convex hull 25
  26. 26. A simple exercise from geometry Exercise: Given a line L and two points p and q, determine whether the points lie on the same/different sides of L. L p q q 26
  27. 27. Conflict graph : a powerful data structure cones 27
  28. 28. Before entering the for loop 28
  29. 29. Before entering the for loop cones 29
  30. 30. 30
  31. 31. 31
  32. 32. cones 32
  33. 33. cones 33
  34. 34. cones 34
  35. 35. cones 35
  36. 36. cones 36
  37. 37. cones 37
  38. 38. cones 38
  39. 39. cones 39
  40. 40. cones 40
  41. 41. 41
  42. 42. 42
  43. 43. 43
  44. 44. 44
  45. 45. 45
  46. 46. Running time of the algorithm 46
  47. 47. 47
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