This document discusses various advanced algorithms and probabilistic analysis techniques. It begins with an overview of probabilistic analysis, which involves making assumptions about the distribution of inputs and analyzing average case running time. It then discusses randomized algorithms, which force all inputs to be equally likely by randomizing the input. As an example, it discusses how randomizing the order of candidates in a hiring problem guarantees the expected number of hires will be n + O(1). It introduces the concept of indicator random variables to convert between probabilities and expectations. It also discusses different types of randomized algorithms like Las Vegas algorithms and Monte Carlo algorithms, providing quicksort as an example of a Las Vegas algorithm.

1. Advanced Algorithms

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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.
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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
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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
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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
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17. What is Expected Value? (IRV)
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• 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
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20. IRV = Probability
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21. IRV Ex. Coin Tossing
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22. IRV ex for Hiring Problem
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23. Expected Value in HIRING
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28. For all n days
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29. Indicator Random variable
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30. Types of randomized algorithms
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33. Las Vegas algorithm
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34. Quick Sort- an example of las Vegas
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36. Monte Carlo
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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)
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