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# 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