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Randomized AlgorithmsCS648

Lecture 7Two applications of Union Theorem• Balls into Bin experiment : Maximum load• Randomized Quick Sort: Concentration of the running

time 1

Union theorem

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Union theorem

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Application 1 of the Union Theorem

balls into Bins: Maximum load

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Balls into Bins

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1 2 3 … i … n

1 2 3 4 5 … m-1 m

Balls into Bins

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1 2 3 … j … n

1 2 3 4 5 … m-1 m

Balls into BinsThe main difficulty and the way out

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1 2 3 … j … n

1 2 3 4 5 … m-1 m

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1 2 3 … j … n

1 2 3 4 5 … m-1 m

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1 2 3 … j … n

1 2 3 4 5 … m-1 m

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Balls into Bins

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Application 2 of the Union Theorem

Randomized Quick sort: The secret of its popularity

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Concentration of Randomized Quick Sort

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A

Concentration of Randomized Quick SortTools needed

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Randomized QuickSort The main difficulty and the way out

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Elements of A arranged in Increasing order of values

Randomized QuickSort The main difficulty and the way out

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Elements of A arranged in Increasing order of values

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Elements of A arranged in Increasing order of values

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Randomized QuickSort A new way to count the comparisons

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Elements of A arranged in Increasing order of values

Randomized QuickSort Applying Union theorem

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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.23

middle-half

Increasing order of values

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Randomized Quick Sort

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middle-half

Increasing order of values

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Randomized Quick Sort

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middle-half

Increasing order of values

…

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Randomized Quick SortFinal result

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Some Well Known and Well STUDIEDRandom Variables

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Bernoulli Random Variable

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Binomial Random Variable

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Geometric Random Variable

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Negative Binomial Random Variable

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