Lecture 10-cs648=2013
- 1. Randomized Algorithms CS648 Lecture 10 Random Sampling part-II (To find a subset with desired property) 1
- 2. Overview • • • • There is a huge list (1 million) of blood donors. Unfortunately the blood group information is missing at present. We need a donor with blood group O+. What to do ? Solution: (Select a random subset of donors.) Repeat until we get a donor of blood group O+. { Pick phone number of a donor randomly uniformly Call him to ask his Blood group. }
- 3. Random Sampling • Suppose there is a computational problem where we require to find a subset with some desired properties. • Unfortunately, computing such a set deterministically may take huge time. • Random sampling carried out suitably may produce a subset with the desired property with some probability.
- 4. RANDOMIZED ALGORITHM FOR BPWM PROBLEM
- 5. Integer Product of Matrices 1 0 0 1 0 1 0 1 0 0 2 1 2 0 0 1 0 0 0 1 0 1 1 0 1 2 0 1 0 0 0 0 1 0 0 1 1 1 1 0 1 1 1 1 0 0 1 0 1 0 1 1 1 0 0 1 2 2 0 1 1 1 0 1 0 1 0 0 1 0 2 2 3 0 D 1 A B
- 6. Boolean Product of Matrices 1 0 0 1 0 1 0 1 0 0 1 1 1 0 0 1 0 0 0 1 0 1 1 0 1 1 0 1 0 0 0 0 1 0 0 1 1 1 1 0 1 1 1 1 0 0 1 0 1 0 1 1 1 0 0 1 1 1 0 1 1 1 0 1 0 1 0 0 1 0 1 1 1 C 0 1 A B
- 7. Boolean Product of Matrices 1 0 0 1 0 1 0 1 0 0 1 1 1 0 0 1 0 0 0 1 0 1 1 0 1 1 0 1 0 0 0 0 1 0 0 1 1 1 1 0 1 1 1 1 0 0 1 0 1 0 1 1 1 0 0 1 1 1 0 1 1 1 0 1 0 1 0 0 1 0 1 1 1 C 0 1 A B
- 8. Boolean Product Witness Matrix (BPWM)
- 10. All Pairs Shortest Paths (APSP)
- 11. All Pairs Shortest Paths (APSP) Students having interest in algorithms are strongly advised to study this novel algorithm from Motwani-Raghwan book or the original journal version. (This is, of course, not part of the syllabus for CS648)
- 13. Boolean Product Witness Matrix (BPWM)
- 14. Boolean Product of Matrices 1 0 0 1 0 1 0 1 0 0 1 1 1 0 0 1 0 0 0 1 0 1 1 0 1 1 0 1 0 0 0 0 1 0 0 1 1 1 1 0 1 1 1 1 0 0 1 0 1 0 1 1 1 0 0 1 1 1 0 1 1 1 0 1 0 1 0 0 1 0 1 1 1 C 0 1 2 1 2 0 0 2 0 1 0 0 1 1 1 1 0 1 2 2 0 1 2 2 3 0 1 A Look carefully at the integer product matrix D. Does it have any thing to do with witnesses. B D
- 15. Boolean Product of Matrices 1 0 0 1 0 1 0 1 0 0 2 1 2 0 0 1 0 0 0 1 0 1 1 0 1 2 0 1 0 0 0 0 1 0 0 1 1 1 1 0 1 1 1 1 0 0 1 0 1 0 1 1 1 0 0 1 2 2 0 1 1 1 0 1 0 1 0 0 1 0 2 2 3 0 D 1 A B
- 16. Boolean Product of Matrices 1 2 3 4 5 ⨯ ⨯ ⨯ ⨯ ⨯ 1 0 0 1 0 1 0 1 0 0 2 1 2 0 0 1 0 0 0 1 0 1 1 0 1 2 0 1 0 0 0 0 1 0 0 1 1 1 1 0 1 1 1 1 0 0 1 0 1 0 1 1 1 0 0 1 2 2 0 1 1 1 0 1 0 1 0 0 1 0 2 2 3 0 D 1 A There is a way to manipulate A so that D will store a witness for all those pairs which have singleton witness. Can you guess ? B
- 17. Boolean Product of Matrices 1 2 3 4 5 ⨯ ⨯ ⨯ ⨯ ⨯ 1 0 0 1 0 1 0 1 0 0 2 4 2 0 0 1 0 0 0 1 0 1 1 0 1 2 0 1 0 0 0 0 1 0 0 1 1 1 1 0 3 3 3 3 0 0 1 0 1 0 1 1 1 0 0 2 2 2 0 2 1 1 0 1 0 1 0 0 1 0 2 2 3 0 D 2 A B
- 18. Boolean Product of Matrices 1 2 3 4 5 ⨯ ⨯ ⨯ ⨯ ⨯ 1 0 0 1 0 1 0 1 0 0 5 4 5 0 0 1 0 0 0 1 0 1 1 0 1 6 0 1 0 0 0 0 1 0 0 1 1 1 1 0 3 3 3 3 0 0 1 0 1 0 1 1 1 0 0 2 6 6 0 2 1 1 0 1 0 1 0 0 1 0 5 6 7 0 D 2 A B
- 19. Algorithm for Computing Singleton Witnesses
- 20. Algorithm Design for BPWM
- 22. 1 2 3 4 … n ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ 1 1 0 1 1 0 1 1 1 0 0 1 0 1 1 0 1 1 0 0 A B
- 23. 1 2 3 4 … n ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ 1 1 0 1 1 0 1 1 1 0 0 1 0 1 1 0 1 1 0 0 A B
- 24. 1 2 3 4 5 … n-1 n ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ 1 1 0 1 1 0 1 1 1 0 0 1 0 1 1 0 1 1 0 0 A B
- 25. 0 2 3 0 5 … n-1 0 ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ 1 1 0 1 1 0 1 1 1 0 0 1 0 1 1 0 1 1 0 0 A B
- 26. No idea! Let us ask the following related but easier question.
- 32. Conclusion A sketch of the solution for Question 1 was given in the class. The students are encouraged to work out the exact details. The solution will be presented in the beginning of next lecture class.