Lecture 9-cs648-2013 Randomized Algorithms

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Lecture 9-cs648-2013 Randomized Algorithms

  1. 1. Randomized Algorithms CS648 Lecture 9 Random Sampling part-I (Approximating a parameter) 1
  2. 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. 3. AN INSPIRATIONAL PROBLEM FROM CONTINUOUS PROBABILITY
  4. 4. 0 1
  5. 5. 0 1 Sampling points on a line segment 0 1
  6. 6. Sampling points on a Circle (of circumference 1) 1
  7. 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. 8. Transforming a line segment to a circle (just a different perspective) First uniformly random point is the knot.
  9. 9. 0 1 We have got the answer of the problem (without any knowledge of continuous probability theory) 0 1
  10. 10. ESTIMATING THE NUMBER OF BALLS IN A BAG
  11. 11. Estimating the number of Balls in a BAG 4 t 1 2 3 5 n j q : c : i l l : : : :: :
  12. 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. 13. Estimating the number of Balls in a BAG 4 t 1 2 3 5 n j q : c : i l l : : : :: :
  14. 14. How good is the estimate ? 2 N1 N-1 multiple sampling.
  15. 15. Multiple samplings to improve accuracy and reduce error probability 21 N
  16. 16. A better algorithm for estimating the number of balls:
  17. 17. 21 N
  18. 18. Final result
  19. 19. Randomized framework for estimating a parameter
  20. 20. ESTIMATING THE SIZE OF TRANSITIVE CLOSURE OF A DIRECTED GRAPH
  21. 21. Estimating size of Transitive Closure of a Directed Graph
  22. 22. Estimating size of Transitive Closure of a Directed Graph
  23. 23. Estimating size of Transitive Closure of a Directed Graph
  24. 24. Randomized Monte Carlo Algorithm for estimating the size of transitive closure of directed graph
  25. 25. MIN-Label Problem
  26. 26. MIN-Label Problem
  27. 27. MIN-Label Problem
  28. 28. Inference from the inspirational problem
  29. 29. RANDOMIZED MONTE CARLO ALGORITHM FOR ESTIMATING THE SIZE OF TRANSITIVE CLOSURE OF A DIRECTED GRAPH
  30. 30. 0.45 0.71 0.22 0.53 0.830.38
  31. 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. 32. Estimating size of Transitive Closure of a Directed Graph
  33. 33. Estimating size of Transitive Closure of a Directed Graph
  34. 34. 0 1 Can you answer Question 2 now ?
  35. 35. Estimating size of Transitive Closure of a Directed Graph
  36. 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.

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