Randomized AlgorithmsCS648

Lecture 20Probabilistic Method

(part 1)

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PROBABILISTIC METHOD

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Probabilistic methods

Methods that use• Probability theory• Randomized algorithm

to prove deterministic combinatorial results

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PROBLEM 1HOW MANY MIN CUTS ?

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Min-Cut: undirected connected graph Definition (cut):A subset whose removal disconnects the graph.

Definition (min-cut): A cut of smallest size.Question: How many cuts can there be in a graph?Question: How many min-cuts can there be in a graph?

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𝟐𝒏−𝟐𝟐

Algorithm for min-cut

Min-cut():{ Repeat times {

Let ; Contract().

} return the edges of multi-graph ;} Running time:

Question: What is the sample space of the output of the algorithm ?Answer: all-cuts of .

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Analysis of Algorithm for min-cut

Let be any arbitrary min-cut.

Question: What is probability that is preserved during the algorithm ?Answer:

= =

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Number of min-cuts

Let there be min-cuts in .Let these min-cuts be .

Define event : “output of the algorithm Min-cut() is ”.P()

P() Surely P()

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𝟐𝒏(𝒏−𝟏)

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PROBLEM 2HOW MANY ACUTE TRIANGLES ?

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How many acute triangles

Problem Definition:There is a set of points in plane and no three of them are collinear. How many triangles formed by these points are acute ?Answer: At most

Solution: Let : probability that a triangle formed by 3 random points from is acute.

Show that

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points

Case 1:

Sum of the four angles is . at least one of them has to be Hence, at least one of the four triangles is non-acute.

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points

Case 2:

Sum of the three angles at the center is . at least two of these angles have to be at least 2 of the four triangles is non-acute.

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points points

Lemma1: A triangle formed by selecting 3 points randomly uniformly from 4 points is acute triangle with probability at most .

Lemma2: A triangle formed by selecting 3 points randomly uniformly from 5 points is acute triangle with probability at most .

(Do it as a simple exercise using Lemma 1.)

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Two stage sampling

: a set of elements. Let be a uniformly random sample of elements from .Let be a uniformly random sample of elements from .

Question: What can we say about (probability distribution of) ?Answer: is a uniformly random sample of elements from .

(Do it as a simple exercise. It uses elementary probability)

Can you use this answer to calculate ?

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Number of acute triangles

: set of points. : probability that a triangle formed by 3 random points from is acute. = ?

: a uniformly random sample of points from . : a uniformly random sample of points from .

= P(a random triangle from is acute) // use previous slide and elementary prob.

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PROBLEM 3SUM FREE SUBSET OF LARGE SIZE

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Large subset that is sum-free

Problem Definition:There is a set of positive integers. Aim is to compute a large subset such that there do not exist three elements , , such that

How large can be for any arbitrary ?

Answer: At least

Spend some time to understand this problem and to realize its difficulty.

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Large subset that is sum-free

Let be a prime number.Let . //The other choice is also fine here.

A randomized algorithm:Select a random number from {}. Map each element to mod . all those elements of that get mapped to {} ?Return ;

Question: What is the expected number of elements from that are mapped to {} ? Answer:

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To prove it, use • the fact that mapping is 1-1

and uniform.• and Linearity of expectation.

Large subset that is sum-free

Let be a prime number.Let .

A randomized algorithm:Select a random number from {}. Map each element to mod . all those elements of that get mapped to {} ?Return ;

Claim: is sum-free.

Try to prove it before going to the next slide

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Showing that is sum-free.

Let and be any two elements in .Let gets mapped to and gets mapped to and ,

Hence and

we just need to show that , if present in , must not be mapped in . will be mapped to ??Give suitable arguments to conclude that• must be greater than .• If , then would be strictly less than .

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1 2 … … … … … 𝜶 𝜷

• Try to ponder over the entire solution given for the Large sum-free subset problem.

• Try to realize the importance of each part of the solution (primality of , the choice of middle third, …)

• This solution is one of those gems of discrete probability / randomized algorithm which you would like to revisit even after this course.

• I just wonder how such a great solution can come to one’s mind…

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PROBLEM 4LARGE CUT IN A GRAPH

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Large cut in a graph

Problem Definition:Let be an undirected graph on vertices and edges. How large can any cut in be ?

Answer: At least

Spend some time to find out a proof for this bound. Hopefully, after 3 problems, you would have realized the way probabilistic method works.

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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 .

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Large cut in a graph

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

E[]

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Large cut in a graph

Now use the following result which is simple but very useful.

Let is a random variable defined over a probability space .If , then there exists an elementary event , such that

Use it to conclude that there is a cut of size at least .

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