Lecture 14-cs648-2013
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This document discusses two randomized algorithms: 1) A distributed client-server problem where clients randomly select servers to send job requests to. It analyzes the expected number of rounds needed to complete all jobs using a two-stage approach. 2) A rumor spreading problem where the number of people knowing a rumor can double each day. It also analyzes this problem using a two-stage approach to calculate the expected number of days needed for everyone to know the rumor.
Lecture 14-cs648-2013
- 1. Randomized Algorithms CS648 Lecture 14 Expected duration of a randomized experiment Part II 1
- 7. Distributed Client-Server Problem Randomized protocol (one round) • Each client sends a request to a server selected randomly uniformly and independently. • Each server which receives one or more requests, accepts only one request and finishes the corresponding job. • Each client, whose job is finished, leaves the system. • The remaining clients repeat the same procedure in next round. Question: what is the expected number of rounds to finish all jobs? 7
- 8. Distributed Client-Server problem Randomized protocol 1 2 3 4 5 6 7 8 7 8 Clients It can be framed as our familiar ball-bin problem. Servers 1 2 3 4 5 6 8
- 9. Distributed Client-Server problem Randomized protocol 1 2 1 2 3 4 5 6 7 8 6 7 8 Balls Bins 3 4 5 9
- 10. Distributed Client-Server problem Randomized protocol Round 1 1 2 1 2 3 3 4 4 5 5 6 7 8 6 7 8 10
- 11. Distributed Client-Server problem Randomized protocol 3 1 2 3 4 4 7 5 6 8 7 8 11
- 12. Distributed Client-Server problem Randomized protocol Round 2 3 1 2 3 4 4 7 5 6 8 7 8 12
- 13. Distributed Client-Server problem Randomized protocol Round 3 8 1 2 3 4 5 6 7 8 13
- 14. CALCULATING EXPECTED NO. OF ROUNDS FIRST APPROACH 14
- 15. Round 1 1 Distributed Client-Server problem Randomized protocol 2 3 Not so easy to find 15
- 16. Round 1 1 Distributed Client-Server problem Randomized protocol 2 3 Is there any relation between no. of empty bins and no. of balls leaving the system in round 1 ? 16
- 17. Distributed Client-Server problem Randomized protocol 1 2 17
- 18. Distributed Client-Server problem Randomized protocol Round 0 No. of balls in the system Fraction of balls in the system 1 1 2 This table gives the intuition for the expected no. of rounds but it directly does not help us to calculate the expected no. of rounds ? It also does not directly help to get a high prob. Bound. Convince yourself before proceeding further. 18
- 19. Distributed Client-Server problem Randomized protocol Use Markov’s Inequality Use Recurrence 1 Each round is good independent of other rounds. 19
- 20. Distributed Client-Server problem Randomized protocol This bound is very loose. Can you see why ? 20
- 21. An important insight that we missed Question: What is the cause of multiple rounds for a ball ? Answer: presence of other competing balls INSIGHT As the algorithm proceeds: • The number of these competing balls reduce • but the number of bins remain unchanged Chances of a ball to leave the system increases as the algorithm proceeds. 21
- 22. CALCULATING EXPECTED NO. OF ROUNDS WITH NEW INSIGHT 22
- 23. Distributed Client-Server problem Randomized protocol Stage 1 Stage 2 23
- 24. CALCULATING EXPECTED NO. OF ROUNDS IN STAGE 2 24
- 25. Distributed Client-Server problem Randomized protocol 1 2 We need a reasonably good upper bound for this expression 25
- 26. Distributed Client-Server problem Randomized protocol 1 2 26
- 27. Distributed Client-Server problem Randomized protocol Use Markov’s Inequality Use Recurrence 2 27
- 28. Distributed Client-Server problem Randomized protocol 28
- 29. Distributed Client-Server problem Randomized protocol 29
- 32. Rumor Spreading Number of people knowing the rumor can only double in a day. 32
- 35. Rumor Spreading Stage 1 Stage 2 35
- 36. EXPECTED NO. OF DAYS IN STAGE 2 36
- 38. EXPECTED NO. OF DAYS IN STAGE 1 38
- 39. It is left as a home work to be discussed in some future class. I shall be happy to see a correct solution by some of you in the next class. 39