1. Randomized Algorithms Randomized Algorithms CS648 Lecture 20 Probabilistic Method (part 1) Lecture 20 Probabilistic Method (part 1) 1

2. PROBABILISTIC METHOD 2

3. Probabilistic methods Methods that use Probability theory Randomized algorithm to prove deterministic combinatorial results 3

4. PROBLEM 1 HOW MANY MIN CUTS ? 4

5. Min-Cut 5

6. Algorithm for min-cut 6

7. Analysis of Algorithm for min-cut 7

8. Number of min-cuts 8

9. PROBLEM 2 HOW MANY ACUTE TRIANGLES ? 9

10. How many acute triangles 10

11. 11

12. 12

13. 13

14. Two stage sampling 14

15. Number of acute triangles 15

16. PROBLEM 3 SUM FREE SUBSET OF LARGE SIZE 16

17. Large subset that is sum-free 17

18. Large subset that is sum-free 18 To prove it, use the fact that mapping is 1-1 and uniform. and Linearity of expectation.

19. Large subset that is sum-free 19

20. 20

21. 21

22. PROBLEM 4 LARGE CUT IN A GRAPH 22

23. Large cut in a graph 23

24. Large cut in a graph 24

25. Large cut in a graph 25

26. Large cut in a graph 26