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Randomized Algorithms Randomized Algorithms CS648 Lecture 9 Random Sampling part-I (Approximating a parameter) Lecture 9 Random Sampling part-I (Approximating.

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Presentation on theme: "Randomized Algorithms Randomized Algorithms CS648 Lecture 9 Random Sampling part-I (Approximating a parameter) Lecture 9 Random Sampling part-I (Approximating."— Presentation transcript:

1 Randomized Algorithms Randomized Algorithms CS648 Lecture 9 Random Sampling part-I (Approximating a parameter) 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

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11 ESTIMATING THE NUMBER OF BALLS IN A BAG

12 Estimating the number of Balls in a BAG 4 t n j q :c:c : i l l : : : :: :

13 4 t n j q :c:c : i l l : : : :: : Can we use it to design an algorithm ?

14 Estimating the number of Balls in a BAG 4 t n j q :c:c : i l l : : : :: :

15 How good is the estimate ? 2 N 1 N-1 multiple sampling.

16 Multiple samplings to improve accuracy and reduce error probability 21N

17 A better algorithm for estimating the number of balls:

18 21N

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

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25 Randomized Monte Carlo Algorithm for estimating the size of transitive closure of directed graph

26 MIN-Label Problem

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29 Inference from the inspirational problem

30 RANDOMIZED MONTE CARLO ALGORITHM FOR ESTIMATING THE SIZE OF TRANSITIVE CLOSURE OF A DIRECTED GRAPH

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34 Estimating size of Transitive Closure of a Directed Graph

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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.


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