Lecture 9-cs648-2013 Randomized Algorithms
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This document discusses random sampling techniques for estimating parameters. It presents an inspirational problem of sampling points on a line segment or circle uniformly at random. This problem is used to motivate an algorithm for estimating the number of balls in a bag by randomly sampling balls with replacement and estimating the total based on the sample proportion. The document then discusses extending this technique to estimate the size of the transitive closure of a directed graph by randomly sampling node pairs and propagating labels to estimate the number of reachable pairs. Homework is assigned to analyze the error bounds of this technique using Chernoff bounds.
Lecture 9-cs648-2013 Randomized Algorithms
- 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
- 11. ESTIMATING THE NUMBER OF BALLS IN A BAG
- 12. Estimating the number of Balls in a BAG 4 t 1 2 3 5 n j q : c : i l l : : : :: :
- 13. 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 ?
- 14. Estimating the number of Balls in a BAG 4 t 1 2 3 5 n j q : c : i l l : : : :: :
- 15. How good is the estimate ? 2 N1 N-1 multiple sampling.
- 16. Multiple samplings to improve accuracy and reduce error probability 21 N
- 17. A better algorithm for estimating the number of balls:
- 18. 21 N
- 19. Final result
- 20. Randomized framework for estimating a parameter
- 21. ESTIMATING THE 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. Estimating size of Transitive Closure of a Directed Graph
- 25. Randomized Monte Carlo Algorithm for estimating the size of transitive closure of directed graph
- 29. Inference from the inspirational problem
- 30. RANDOMIZED MONTE CARLO ALGORITHM FOR ESTIMATING THE SIZE OF TRANSITIVE CLOSURE OF A DIRECTED GRAPH
- 34. Estimating size of Transitive Closure of a Directed Graph
- 35. Estimating size of Transitive Closure of a Directed Graph
- 36. 0 1 Can you answer Question 2 now ?
- 37. Estimating size of Transitive Closure of a Directed Graph
- 38. 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.