Randomized Algorithms CS648

1. Randomized AlgorithmsCS648 Lecture 20 Probabilistic Method (part 1)

2. Probabilistic method

3. Probabilistic methods Methods that use • Probability theory • Randomized algorithm to prove deterministiccombinatorial results

4. problem 1How Many min CUTs ?

5. 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?

6. 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 .

7. Analysis of Algorithm for min-cut Let be any arbitrary min-cut. Question: What is probability that is preserved during the algorithm ? Answer: = =

8. 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() 

9. problem 2How many Acute Triangles ?

10. 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

11. 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.

12. 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.

13. 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.)

14. 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 ?

15. 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. 

16. problem 3Sum free subset of large size

17. 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.

18. 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.

19. 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 

20. 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 … … … … …

21. 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…

22. problem 4Large CUT in a graph

23. 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.

24. Large cut in a graph A randomized algorithm: ∅; Add each vertex from to randomly independently with probability . Return the cut defined by .

25. Large cut in a graph : size of cut () returned by the randomized algorithm. E[] = ?? • E[]

26. 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 .