Randomized AlgorithmsCS648

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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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 QuickSortThe main difficulty and the way out

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Elements of A arranged in Increasing order of values

Randomized QuickSortThe 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 QuickSortA new way to count the comparisons

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Elements of A arranged in Increasing order of values

Randomized QuickSortApplying 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

…

Randomized Quick Sort

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middle-half

Increasing order of values

…

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 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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