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Randomized Algorithms CS648. Lecture 20 Probabilistic Method (part 1). Probabilistic method . Probabilistic methods . Methods that use Probability theory Randomized algorithm t o prove deterministic combinatorial results. problem 1 How Many min CUTs ? . Min-Cut.

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randomized algorithms cs648

Randomized AlgorithmsCS648

Lecture 20

Probabilistic Method

(part 1)

probabilistic methods
Probabilistic methods

Methods that use

  • Probability theory
  • Randomized algorithm

to prove deterministiccombinatorial results

min cut
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?

algorithm for min cut
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 .

analysis of algorithm for min cut
Analysis of Algorithm for min-cut

Let be any arbitrary min-cut.

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

Answer:

=

=

number of min cuts
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()

how many acute triangles
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

points
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.

points1
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.

points points
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.)

two stage sampling
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 ?

number of acute triangles
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.

large subset that is sum free
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.

large subset that is sum free1
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:

  • To prove it, use
  • the fact that mapping is 1-1 and uniform.
  • and Linearity of expectation.
large subset that is sum free2
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 

showing that is sum free
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 .

1 2 … … … … …

slide21

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…
large cut in a graph
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.

large cut in a graph1
Large cut in a graph

A randomized algorithm:

∅;

Add each vertex from to randomly independently with probability .

Return the cut defined by .

large cut in a graph2
Large cut in a graph

: size of cut () returned by the randomized algorithm.

E[] = ??

  • E[]
large cut in a graph3
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 .