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# Lecture 9-cs648-2013 Randomized Algorithms

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• 1. Randomized Algorithms CS648 Lecture 9 Random Sampling part-I (Approximating a parameter) 1
• 2. Overview of the Lecture Randomization Framework for estimation of a parameter 1. Number of balls from a bag 2. Size of transitive closure of a directed graph • An Inspirational Problem from Continuous probability
• 3. AN INSPIRATIONAL PROBLEM FROM CONTINUOUS PROBABILITY
• 4. 0 1
• 5. 0 1 Sampling points on a line segment 0 1
• 6. Sampling points on a Circle (of circumference 1) 1
• 7. Transforming a line segment to a circle (just a different perspective) The knot formed by joining the ends of the line segment Give the knot a uniformly random rotation around the circle
• 8. Transforming a line segment to a circle (just a different perspective) First uniformly random point is the knot.
• 9. 0 1 We have got the answer of the problem (without any knowledge of continuous probability theory) 0 1
• 10. ESTIMATING THE NUMBER OF BALLS IN A BAG
• 11. Estimating the number of Balls in a BAG 4 t 1 2 3 5 n j q : c : i l l : : : :: :
• 12. Estimating the number of Balls in a BAG 4 t 1 2 3 5 n j q : c : i l l : : : :: : Can we use it to design an algorithm ?
• 13. Estimating the number of Balls in a BAG 4 t 1 2 3 5 n j q : c : i l l : : : :: :
• 14. How good is the estimate ? 2 N1 N-1 multiple sampling.
• 15. Multiple samplings to improve accuracy and reduce error probability 21 N
• 16. A better algorithm for estimating the number of balls:
• 17. 21 N
• 18. Final result
• 19. Randomized framework for estimating a parameter
• 20. ESTIMATING THE SIZE OF TRANSITIVE CLOSURE OF A DIRECTED GRAPH
• 21. Estimating size of Transitive Closure of a Directed Graph
• 22. Estimating size of Transitive Closure of a Directed Graph
• 23. Estimating size of Transitive Closure of a Directed Graph
• 24. Randomized Monte Carlo Algorithm for estimating the size of transitive closure of directed graph
• 25. MIN-Label Problem
• 26. MIN-Label Problem
• 27. MIN-Label Problem
• 28. Inference from the inspirational problem
• 29. RANDOMIZED MONTE CARLO ALGORITHM FOR ESTIMATING THE SIZE OF TRANSITIVE CLOSURE OF A DIRECTED GRAPH
• 30. 0.45 0.71 0.22 0.53 0.830.38
• 31. 0.34 0.14 0.45 0.71 0.22 0.53 0.83 0.28 0.901 0.65 0.265 0.49 0.54 0.74 0.38 0.81 0.63
• 32. Estimating size of Transitive Closure of a Directed Graph
• 33. Estimating size of Transitive Closure of a Directed Graph
• 34. 0 1 Can you answer Question 2 now ?
• 35. Estimating size of Transitive Closure of a Directed Graph
• 36. Homework Use Chernoff bound to get a high probability bound on the error. Hint: Proceed along similar lines as in the case of estimating number of balls in a bag. Make sincere attempts to do this homework. I shall discuss the same briefly in the beginning of the next class.