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Randomized Algorithms
CS648

Lecture 16
Randomized Incremental Construction
(Backward analysis)
1
PROBLEM 1
FIND-MIN PROBLEM

2
Find-Min algorithm
A
1

2

…

??

3
A
1

2

…

4
5
A
1

2

…

6
Fact: A is permuted randomly uniformly

7
A
1

2

…

8
PROBLEM 2
CLOSEST PAIR OF POINTS

9
Closest Pair of Points

10
11
O(1)
O(1)

O(1)

12
13
14
15
16
17
18
RANDOMIZED INCREMENTAL
CONSTRUCTION

19
Randomized Incremental Construction

20
Randomized Incremental Construction
• Convex Hull of a set of points
• Trapezoidal decomposition of a set of segments.
• C...
PROBLEM 3
CONVEX HULL OF POINTS

22
Convex hull of Points

23
Convex hull of Points

24
Randomized algorithm for convex hull

25
A simple exercise from geometry
Exercise: Given a line L and two points p and q, determine whether the
points lie on the s...
Conflict graph : a powerful data structure
cones

27
Before entering the for loop

28
Before entering the for loop
cones

29
30
31
cones

32
cones

33
cones

34
cones

35
cones

36
cones

37
cones

38
cones

39
cones

40
41
42
43
44
45
Running time of the algorithm

46
47
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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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