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Randomized Algorithms Randomized Algorithms CS648 Lecture 3 Two fundamental problems Balls into bins Randomized Quick Sort Random Variable and Expected value Lecture 3 Two fundamental problems Balls into bins Randomized Quick Sort Random Variable and Expected value 1

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BALLS INTO BINS CALCULATING PROBABILITY OF SOME INTERESTING EVENTS 2

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Balls into Bins 3 1 2 3 … i … n 1 2 3 4 5 … m-1 m

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Balls into Bins Question : What is the probability that there is at least one empty bin ? 4 1 2 3 … i … n 1 2 3 4 5 … m-1 m

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Balls into Bins 5 1 2 3 … i … n 1 2 3 4 5 … m-1 m

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Balls into Bins 6 1 2 3 … i … n 1 2 3 4 5 … j … m-1 m disjoint Independent

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Balls into Bins 7 1 2 3 … i … n 1 2 3 4 5 … j … m-1 m

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Balls into Bins 8 1 2 3 … i … n 1 2 3 4 5 … j … m-1 m

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Balls into Bins 9 Express the event as union of some events …

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Balls into Bins 10

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Balls into Bins 11

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RANDOMIZED QUICK SORT WHAT IS PROBABILITY OF TWO SPECIFIC ELEMENTS GETTING COMPARED ? 12

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Randomized Quick Sort 13

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Randomized Quick Sort 14 Not a feasible way to calculate the probability

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15 Go through the next few slides slowly and patiently, pondering at each step. Never accept anything until and unless you can see the underlying truth yourself.

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16 Elements of A arranged in Increasing order of values

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ALTERNATE SOLUTION USING ANALOGY TO ANOTHER RANDOM EXPERIMENT Remember we took a similar approach earlier too: we used a coin toss experiment to analyze failure probability of Rand- Approx-Median algorithm. 20

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A Random Experiment: A Story of two friends 21

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Viewing the entire experiment from perspective of A and B 22 1 2 3 4 … n-1 n AB

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Viewing the entire experiment from perspective of A and B 23 1 2 3 4 … n-1 n AB

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Viewing the entire experiment from perspective of A and B 24 AB

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Viewing the entire experiment from perspective of A and B 25 AB

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Viewing the entire experiment from perspective of A and B 26 AB

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Viewing the entire experiment from perspective of A and B 27 AB

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PROBABILITY THEORY (RANDOM VARIABLE AND EXPECTED VALUE) 28

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Random variable 29 Randomized- Quick-Sort on array of size n Number of HEADS in 5 tosses Sum of numbers in 4 throws Number of comparisons

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Random variable 30

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Many Random Variables for the same Probability space Random Experiment: Throwing a dice two times X : the largest number seen Y : sum of the two numbers seen 31

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Expected Value of a random variable (average value) 32 Ω X= a X= b X= c

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Examples 33

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Can we solve these problems ? 34 Spend at least half an hour to solve these two problems using the tools you know. This will help you appreciate the very important concept we shall discuss in the next class.

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