Lecture 4-cs648 Randomized Algorithms
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This document summarizes key points from a lecture on randomized algorithms: 1. It introduces linearity of expectation as the most important tool for analyzing randomized algorithms. 2. It discusses random variables and expected value, and gives examples of analyzing the number of empty bins from balls being placed in bins randomly and the number of comparisons in randomized quicksort. 3. It poses the main questions these examples aim to answer using linearity of expectation and introduces this principle as well as where and how it can be used.
Lecture 4-cs648 Randomized Algorithms
- 1. Randomized Algorithms CS648 Lecture 4 • Linearity of Expectation with applications (Most important tool for analyzing randomized algorithms) 1
- 2. RECAP FROM THE LAST LECTURE 2
- 4. Expected Value of a random variable (average value) 4 Ω X= a X= b X= c
- 5. Examples 5
- 6. Can we solve these problems ? 6
- 7. Balls into Bins (number of empty bins) 7 1 2 3 … … n 1 2 3 4 5 … m-1 m This is a right but useless answer !
- 8. Randomized Quick Sort (number of comparisons) 8 We can not proceed from this point … A recursion tree associated with Randomized Quick Sort
- 9. 9 1 2 3 4 5 … m-1 m 1 2 3 … … n Balls into Bins (number of empty bins) Randomized Quick Sort (number of comparisons)
- 10. Balls into Bins (number of empty bins) 10 1 2 3 4 5 … m-1 m
- 11. 11 1 2 3 4 5 1 2 3 4 5 6 0 1 0 1 0
- 12. Sum of Random Variables 12
- 13. Randomized Quick Sort (number of comparisons) 13 Elements of A arranged in Increasing order of values
- 14. 14 1 0 … 0 1 1 … 0
- 15. What have we learnt till now? 15
- 16. The main question ? 16
- 17. Balls into Bins (number of empty bins) 17 1 2 3 … … n 1 2 3 4 5 … m-1 m
- 18. Randomized Quick Sort (number of comparisons) 18
- 20. Where to use Linearity of expectation ? 20
- 21. Think over the following questions? 21
- 22. Think over the following questions? 22
- 24. Some Practice problems as homework • Balls into bin problem: • What is the expected number of bins having exactly 2 balls ? • We toss a coin n times, what is the expected number of times pattern HHT appear ? • A stick has n joints. The stick is dropped on floor and in this process each joint may break with probability p independent of others. As a result the stick will be break into many substicks. – What is the expected number of substicks of length 3 ? – What is the expected number of all the substicks ? 24
- 25. PROBLEMS OF THE NEXT LECTURE 25