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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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