1. Randomized Algorithms
CS648
Lecture 7
Two applications of Union Theorem
• Balls into Bin experiment : Maximum load
• Randomized Quick Sort: Concentration of the running time
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2. Union theorem
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3. Union theorem
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4. APPLICATION 1 OF THE UNION THEOREM
BALLS INTO BINS: MAXIMUM LOAD
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5. Balls into Bins
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1 2 3 … i … n
1 2 3 4 5 … m-1 m

6. Balls into Bins
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1 2 3 … j … n
1 2 3 4 5 … m-1 m

7. Balls into Bins
The main difficulty and the way out
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1 2 3 … j … n
1 2 3 4 5 … m-1 m

8. 8
1 2 3 … j … n
1 2 3 4 5 … m-1 m

9. 9
1 2 3 … j … n
1 2 3 4 5 … m-1 m

10. 10

11. 11

12. Balls into Bins
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13. APPLICATION 2 OF THE UNION THEOREM
RANDOMIZED QUICK SORT:
THE SECRET OF ITS POPULARITY
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14. Concentration of Randomized Quick Sort
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A

15. Concentration of Randomized Quick Sort
Tools needed
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16. Randomized QuickSort
The main difficulty and the way out
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Elements of A arranged in
Increasing order of values

17. Randomized QuickSort
The main difficulty and the way out
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Elements of A arranged in
Increasing order of values

18. 18
Elements of A arranged in
Increasing order of values

19. 19

20. Randomized QuickSort
A new way to count the comparisons
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Elements of A arranged in
Increasing order of values

21. Randomized QuickSort
Applying Union theorem
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22. 22

23. Randomized Quick Sort
Definition: a recursive call is good if the pivot is selected from the middle
half, and bad otherwise.
P(a recursive call is good) = ??
Notation: The size of a recursive call is the size of the subarray it sorts.
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middle-half
Increasing order of values
…

24. Randomized Quick Sort
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middle-half
Increasing order of values
…

25. Randomized Quick Sort
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middle-half
Increasing order of values
…

26. 26

27. Randomized Quick Sort
Final result
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28. SOME WELL KNOWN AND WELL STUDIED
RANDOM VARIABLES
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29. Bernoulli Random Variable
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30. Binomial Random Variable
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31. Geometric Random Variable
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32. Negative Binomial Random Variable
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