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.