This document discusses randomized algorithms and covers several topics: - Chebyshev's Inequality and the Method of Bounded Difference (MOBD) for bounding the probability of deviation of a random variable from its expected value. - MOBD is described as the most powerful method for bounding this probability. - Three examples are presented to illustrate MOBD: balls into bins to calculate empty bins, red-blue balls drawn from a bin, and the number of triangles in a random graph.

1. Randomized Algorithms
CS648
Lecture 22
• Chebyshev Inequality
• Method of Bounded Difference
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2. Chernoff Bound

3. THREE EXAMPLES TO ILLUSTRATE THE
INAPPLICABILITY OF CHERNOFF BOUND
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4. Red-blue balls out of bin
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5. Balls into Bins
(number of empty bins)
1 2 3 4 5
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2
3
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m-1 m
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n
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6. Number of Triangles in a random graph
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7. CHEBYSHEV’S INEQUALITY
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8. Chebyshev’s inequality
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9. Chebyshev’s inequality
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10. METHOD OF BOUNDED DIFFERENCE (MOBD)
The most powerful method for bounding
the probability of deviation of a random variable from expected value
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11. The Power of MOBD
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12. Notations
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13. A new perspective
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The next slide will give a visual description of the process
mentioned above.
But ponder over this slide before pressing the next button.
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17. THE INTUITION
UNDERLYING MOBD
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18. Method of Bounded Difference
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19. Method of Bounded Difference
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20. Method of Bounded Difference - I
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21. Method of Bounded Difference - II
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22. MOBD SUBSUMES CHERNOFF BOUND
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23. MOBD subsumes Chernoff Bound
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24. PROBLEM 1
NO. OF EMPTY BINS
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25. Balls into Bins
(number of empty bins)
1 2 3 4 5
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2
3
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n-1 n
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n
This will give very inferior bound
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26. Balls into Bins
(number of empty bins)
1 2 3 4 5
1
2
3
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n-1 n
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n
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27. Balls into Bins
(number of empty bins)
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28. PROBLEM 2
RED-BLUE BALLS OUT OF BIN
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29. Red-blue balls out of bin
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30. Red-blue balls out of bin
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31. Red-blue balls out of bin
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32. PROBLEM 3
NO. OF TRIANGLES IN RANDOM GRAPH
Do it as exercise.
This problem will also be posted in practice sheet.
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