Randomized AlgorithmsCS648

Lecture 24• Random bit complexity • Derandomization

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Random bit complexity

Definition : The total number of random bits taken from the Random Bit Generator by the algorithm is called its Random bit complexity.

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Random Bit generator

A Randomized Algorithm(for Min-Cut, QuickSort, RIC,…)

Input

RECALL THE NOTION OF INDEPENDENCE

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Types of independences

Definition: are said to be mutually independent if

Definition: are said to be pairwise independent if for every

∙

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Types of independences

Definition: are said to be mutually independent random variables if for any

Definition: are said to be pairwise independent random variables if for every and every

∙

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Important facts

A randomized algorithm typically require random bits/numbers that have• a uniform distribution• pairwise independence

Random bit complexity can be reduced.

Theorem: We can generate pairwise independent random bits usingOnly mutually independent random bits.

We shall now prove this theorem.

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GENERATING UNIFORMLY RANDOM AND PAIRWISE INDEPENDENT BITS

using few truly random bits

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Generating Uniformly Random and pairwise independent Bits

Let be mutually independent random bits.Aim: To generate pairwise independent random bits Key idea: Generate all non-empty subsets of {}Ex:

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0 0 0

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

1 1 1

{ }

{ }

{ }

{ }

{ , }

{ , }{ , , }

2 1 0

𝑋 0

𝑋 1

𝑋 2

𝑋 1⊕𝑋 0

𝑋 2⊕𝑋 0

𝑋 2⊕𝑋 1

𝑋 2⊕𝑋 1⊕𝑋 0

Why the XOR operation ? You should get its answer yourself after a few

slides…

Generating Uniformly Random and pairwise independent Bits

Let be mutually independent random bits.Aim: To generate pairwise independent random bits

Algorithm:For to {

Consider binary representation of ;Let the bits at places only (in this representation) are ; ∙∙∙

}

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Generating Uniformly Random and pairwise independent Bits

, Lemma: Each is a uniformly random bit.Proof: Let = ∙∙∙

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¿𝟏𝟐

Generating Uniformly Random and pairwise independent Bits

, Lemma: ’s are pairwise independent.Proof: Let = ∙∙∙ and = ∙∙∙{, ,…, } ≠ {, ,…, } Without loss of generality, let {, ,…, }Let .

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𝟏𝟐

𝟏𝟐

¿𝟏𝟒

DERANDOMIZATION

transforming a randomized algorithm into a deterministic algorithm

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Large cut in a graph

Theorem: (We proved in Lecture 20)Let be an undirected graph on vertices and edges. There exists a cut of size at least .

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Large cut in a graph

A randomized algorithm: ∅; Add each vertex from to randomly independently with probability . Return the cut defined by .

: size of cut () returned by the randomized algorithm.E[] =

There exists an such that

Question: What is the underlying sample space ?Answer: Depends upon the random bits used by the algorithm.

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Large cut in a graph

Question: How to de-randomize the algorithm ?Answer: Compute cut associated with each and return the largest.Question: How many random bits does the algorithm require ?Answer:

Question: If we use mutually independent bits for all vertices, what is the size of ?Answer: .

Question: Do we really need mutually independent bits for all vertices ?Answer: NO

IDEA : Use only pairwise independent random bits.But will it still ensure E[] = ? Let us see …

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Large cut in a graph

: the pairwise independent random variable for each vertex.: size of cut () returned by the randomized algorithm.E[] = ??

E[]

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¿𝒎𝟐

𝟏𝟒𝟏𝟒

Large cut in a graph

Lemma: If we use only pairwise independent random bits, the expected size of cut will be at least

Question: How many random bits does the algorithm require now ?Answer:

Question: What is the size of now ?Answer: O().

Deterministic algorithm:Just enumerate cuts associated with each and report the largest one.Running time: O()

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Large cut in a graph

Theroem: There is an O() time deterministic algorithm that computes a cut of size at least in a graph having edges and vertices.

In the next class we shall discuss a powerful technique called “Method of Conditional Expectation” to design a O() time algorithm for computing a cut of size at least .

We shall conclude this course with a beautiful puzzle.

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