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 transcript:

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

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

AN INSPIRATIONAL PROBLEM FROM CONTINUOUS PROBABILITY

0 1

0 1 Sampling points on a line segment 0 1

Sampling points on a Circle (of circumference 1) 1

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

Transforming a line segment to a circle (just a different perspective) First uniformly random point is the knot.

0 1 We have got the answer of the problem (without any knowledge of continuous probability theory) 0 1

ESTIMATING THE NUMBER OF BALLS IN A BAG

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

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

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

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

Multiple samplings to improve accuracy and reduce error probability 21N

A better algorithm for estimating the number of balls:

21N

Final result

Randomized framework for estimating a parameter

ESTIMATING THE SIZE OF TRANSITIVE CLOSURE OF A DIRECTED GRAPH

Estimating size of Transitive Closure of a Directed Graph

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

MIN-Label Problem

Inference from the inspirational problem

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

Estimating size of Transitive Closure of a Directed Graph

0 1 Can you answer Question 2 now ?

Estimating size of Transitive Closure of a Directed Graph

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.