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Randomized Algorithms CS648

Randomized Algorithms CS648. Lecture 10 Random Sampling part-II (To find a subset with desired property). 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 ?

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Randomized Algorithms CS648

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  1. Randomized AlgorithmsCS648 Lecture 10 Random Sampling part-II (To find a subset with desired property)

  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. IntegerProduct of Matrices D A B

  6. Boolean Productof Matrices C A B

  7. Boolean Productof Matrices C A B Definition:An integer is said to be witness for a pair () if

  8. Boolean Product Witness Matrix (BPWM) Problem: Given two Boolean matrices A and B, and their Boolean product C, compute a matrix , such that: stores a witness for each () with .

  9. Motivation for BPWM

  10. All Pairs Shortest Paths (APSP) Standard Algorithms: • Floyd WarshalAlgorithm • Dijkstra’s Algorithm • BFS traversal (for unweighted graphs) RaimundSeidel: On the All-Pairs-Shortest-Path Problem in Unweighted Undirected Graphs, JCSS, 51(3): 400-403 (1995) Time complexity for APSP: O time. where is the exponent for multiplying two matrices, currently<2.317 O time in the worst case

  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) Algorithm of RaimundSeidel: • Compute Distance Matrix in O time [Deterministic Algorithm] • Computing Shortest Path Matrix required solving BPWMproblem. • Solving BPWMproblem in Otime [Randomized algorithm]

  12. Randomized algorithm for BPWM

  13. Boolean Product Witness Matrix (BPWM) Problem: Given two Boolean matrices A and B, and their Boolean product C, compute a matrix , such that: stores a witness for each () with . Observations: • There may be many witnesses for a pair () . But our aim is to compute just one witness for each pair. • Witness for any pair () can be searched in O() time. • Verifying if is a witness for a pair () ? (these observations will be used later on. So internalize them fully now.) O() time Just check whether.

  14. Boolean Productof Matrices C A B Look carefully at the integer product matrix D. Does it have any thing to do with witnesses. =number of witnesses for () So it is worth studying for our problem D

  15. Boolean Productof Matrices D A B

  16. Boolean Productof Matrices 1 2 3 4 5 ⨯ ⨯ ⨯ ⨯ ⨯ D A B 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 ?

  17. Boolean Productof Matrices 1 2 3 4 5 ⨯ ⨯ ⨯ ⨯ ⨯ D A B

  18. Boolean Productof Matrices 1 2 3 4 5 ⨯ ⨯ ⨯ ⨯ ⨯ D A B • For pairs having one witness (Yellow entries), stores the witness . • For pairs having multiple witnesses (Blue entries), stores some junk value .

  19. Algorithm for Computing Singleton Witnesses Compute-Singleton-Witnesses(,) { For each  ∙;  ∙; Foreach If is a witness for ()  } Time complexity: O Note that the algorithm is a deterministic one.

  20. Algorithm Design for BPWM Subproblem: How to compute witnesses for all those pairs which have witnesses ? Solution: A randomized Monte Carlo algorithm withO time. Main Problem: How to find witnesses for all pairs ? Solution: A randomized Las Vegas algorithm with expected O time.

  21. RandomizedOAlgorithm:Finding witness for all those pairs which have witnesses

  22. Focus on a single pair (,) 1 2 3 4 … n ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ B A How to nullify the effect of witnesses ? Witnesses for (,)

  23. Focus on a single pair (,) 1 2 3 4 … n ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ B A

  24. Focus on a single pair (,) 1 2 3 4 5 … n-1 n ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ B A

  25. Focus on a single pair (,) 0 2 3 0 5 … n-1 0 ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯ B A will store this witness for (,) now .

  26. Focus on a single pair (,) Question: How to select columns such that exactly one out of witnesses for (,) survives ? Answer: No efficient deterministic algorithm. Idea: (Random sampling) Question:What should be the sampling probability ? Question:What should be the sampling probability such that the expected number of surviving witnesses for (,) turn out to be 1? Answer: Try for the random sampling. No idea! Let us ask the following related but easier question.

  27. Focus on a single pair (,) Question: If each column is selected independently with , what is the probability that exactly one out of witnesses for (,) survives ? Answer: = =

  28. Randomized algorithm for Computing Witnesses for all pairs with witnesses //The pseudo code for sampling the (indices of) columns Sample(,) {  ; For each do: add to with probability ; return ; }

  29. Randomized algorithm for Computing Witnesses for all pairs with witnesses Compute-Witnesses(,,) { { Sample(,); For each if then ∙; else 0;  ∙; Foreach If is a witness for ()  } } Time complexity: O Probability of failing to find a witnessfor a single pair (): ?? How to reduce the error probability to ? Repeat the entire process times.

  30. Randomized algorithm for Computing Witnesses for all pairs with witnesses Compute-Witnesses(,,) { Repeat times { Sample(,); For each if then ∙; else 0;  ∙; Foreach If is a witness for ()  } } Time complexity: O Probability of failing to find a witnessfor a single pair (): ??

  31. Randomized algorithm for Computing Witnesses for all pairs with witnesses Compute-Witnesses(,,) { Repeat times { Sample(,); For each if then ∙; else 0;  ∙; Foreach If is a witness for ()  } } Time complexity: O Probability of failing to find a witnessfor any pair having witnesses: ?? Let there be pairs that have exactly witnesses. Apply Union theorem … <

  32. Conclusion Theorem: Given two Boolean matrices and, and integer , there is a randomized Monte Carlo algorithm to compute witnesses for all those pairs which have witnesses. • The running time is O. • The error probability is at most Questions: • How to compute witnesses for all pairs inO time. • How to convert Monte Carlo to Las Vegas ? 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.

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