Probabilistic and Randomized Algorithm

1. Advanced Algorithms

2. RCPIT 2

3. RCPIT 3

4. RCPIT 4

5. RCPIT 5

6. RCPIT 6

7. RCPIT 7

8. RCPIT 8

9. RCPIT 9

10. RCPIT 10

11. RCPIT 11

12. RCPIT 12

13. Probabilistic Analysis
• To use Probabilistic Analysis, we need to understand the data and make certain
assumptions about the distribution of inputs in the data.
• We need to proceed with analysis and computation of average case running time but we
also need to manage the cost of hiring
• So let's start by giving certain priority or rank to the candidate
• We need to assume that candidates might appear in a random manner, which will lead to
uniform random permutation.
• So how many permutations we can have (n!)
• And each permutation have an equal probability to appear as a sequence for the interview.
• So, probabilistic analysis just works on the assumption that the candidates appear in a
random order of their rank or priority, but what if they are not.
• We can proceed with randomized algorithm to bring about more control.
13

14. Randomized Algorithm
• In average-case analysis, we often assume that all inputs are equally
likely
• Actuality, some inputs might be much more likely
• If we're really unlucky, the most likely inputs can be the most costly (as in
some implementations of quicksort)
• What can we do?
• Force all inputs to be equally likely, by randomizing the input
RCPIT 14

15. Randomization for Hiring Problem
• In the hire-assistant problem, we can first randomly permute the lists of
candidates, and then run the algorithm
• Then, for any input, we'd be guaranteed that the expected number of hires
would be in n + O(1)
• How can we randomly permute a list, so that every permutation is equally
as likely?
• That is, how can we shuffle a list, so that every permutation is equally
likely? Assume that we have a good random number generator.
• When the input is random, we refer to the running time as expected running
time
15

16. Indicator Random Variable
• It is method to convert between probabilities and expectation
• Indicator variable associated with event A:
• I{A} = 1, if A occurs
• I{A} = 0, if A does not occur
Example: Flip a coin: Y is a random variable representing the coin flip
• X = I{Y = H} = 1, if Y = Head
• X = I{Y = T} = 0, if Y = Tail
RCPIT 16

17. What is Expected Value? (IRV)
RCPIT 17
• Expected value E[] of a random variable
• Value you "expect" a random variable to have
• Average (mean) value of the variable over many trials
• Does not have to equal the value of any particular trial

18. Expected Value in IRV
RCPIT 18

19. RCPIT 19

20. IRV = Probability
RCPIT 20

21. IRV Ex. Coin Tossing
RCPIT 21

22. IRV ex for Hiring Problem
RCPIT 22

23. Expected Value in HIRING
RCPIT 23

24. RCPIT 24

25. RCPIT 25

26. RCPIT 26

27. RCPIT 27

28. For all n days
RCPIT 28

29. Indicator Random variable
RCPIT 29

30. Types of randomized algorithms
RCPIT 30

31. RCPIT 31

32. RCPIT 32

33. Las Vegas algorithm
RCPIT 33

34. Quick Sort- an example of las Vegas
RCPIT 34

35. RCPIT 35

36. Monte Carlo
RCPIT 36

37. Game theoretic randomized algorithm techniques (Min-Max) Technique
• It Follows the Backtracking Algo.
• Best Move Strategy is used.
• Max will try to Maximize its Utility (Best Move)
• Min will try to Minimize its Utility (Worst Move)
RCPIT 37