Lecture 20-cs648
-
Upload
rajiv-omar -
Category
Technology
-
view
41 -
download
0
Transcript of Lecture 20-cs648
Randomized AlgorithmsCS648
Lecture 20Probabilistic Method
(part 1)
1
PROBABILISTIC METHOD
2
Probabilistic methods
Methods that use• Probability theory• Randomized algorithm
to prove deterministic combinatorial results
3
PROBLEM 1HOW MANY MIN CUTS ?
4
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?
5
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 .
6
Analysis of Algorithm for min-cut
Let be any arbitrary min-cut.
Question: What is probability that is preserved during the algorithm ?Answer:
= =
7
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()
8
𝟐𝒏(𝒏−𝟏)
¿
PROBLEM 2HOW MANY ACUTE TRIANGLES ?
9
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
10
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.
11
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.
12
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.)
13
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 ?
14
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.
15
PROBLEM 3SUM FREE SUBSET OF LARGE SIZE
16
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.
17
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:
18
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
19
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 .
20
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…
21
PROBLEM 4LARGE CUT IN A GRAPH
22
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.
23
Large cut in a graph
A randomized algorithm:∅;Add each vertex from to randomly independently with probability . Return the cut defined by .
24
Large cut in a graph
: size of cut () returned by the randomized algorithm.E[] = ??
E[]
25
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 .
26