Randomized Algorithms CS 648 Lecture 22 Chebyshev Inequality

Randomized Algorithms CS 648 Lecture 22 • Chebyshev Inequality • Method of Bounded Difference 1

Chernoff Bound •

THREE EXAMPLES TO ILLUSTRATE THE INAPPLICABILITY OF CHERNOFF BOUND 3

Red-blue balls out of bin • 4

Balls into Bins (number of empty bins) 1 2 3 4 5 • 1 2 3 … … m-1 m … n 5

Number of Triangles in a random graph • 6

CHEBYSHEV’S INEQUALITY 7

Chebyshev’s inequality • 8

Chebyshev’s inequality • 9

METHOD OF BOUNDED DIFFERENCE (MOBD) The most powerful method for bounding the probability of deviation of a random variable from expected value 10

The Power of MOBD • 11

Notations • 12

A new perspective • … The next slide will give a visual description of the process mentioned above. But ponder over this slide before pressing the next button. 13

THE INTUITION UNDERLYING MOBD 17

Method of Bounded Difference … … … 18

Method of Bounded Difference … … • … 19

Method of Bounded Difference - I … … • … 20

Method of Bounded Difference - II • 21

MOBD SUBSUMES CHERNOFF BOUND 22

MOBD subsumes Chernoff Bound • 23

PROBLEM 1 NO. OF EMPTY BINS 24

Balls into Bins (number of empty bins) 1 2 3 4 5 • 1 2 3 … … n-1 n … n This will give very inferior bound 25

Balls into Bins (number of empty bins) 1 2 3 4 5 • 1 2 3 … … n-1 n … n 26

Balls into Bins (number of empty bins) • 27

PROBLEM 2 RED-BLUE BALLS OUT OF BIN 28

Red-blue balls out of bin • 29

Red-blue balls out of bin • … … … 30

Red-blue balls out of bin • 31

PROBLEM 3 NO. OF TRIANGLES IN RANDOM GRAPH Do it as exercise. This problem will also be posted in practice sheet. 32