randomized algorithms cs648
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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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