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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Union theorem
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2
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 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
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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 Sort Tools 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.
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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 Sort
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middle-half
Increasing order of values
…
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Randomized Quick Sort Final result
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SOME WELL KNOWN AND WELL STUDIED
RANDOM 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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