Lecture 2-cs648

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Lecture 2-cs648

  1. 1. Randomized Algorithms CS648 Lecture 2 • Randomized Algorithm for Approximate Median • Elementary Probability theory 1
  2. 2. RANDOMIZED MONTE CARLO ALGORITHM FOR APPROXIMATE MEDIAN This lecture was delivered at slow pace and its flavor was that of a tutorial. Reason: To show that designing and analyzing a randomized algorithm demands right insight and just elementary probability. 2
  3. 3. A simple probability exercise 3
  4. 4. 4
  5. 5. Approximate median Definition: Given an array A[] storing n numbers and ϵ > 0, compute an element whose rank is in the range [(1- ϵ)n/2, (1+ ϵ)n/2]. Best Deterministic Algorithm: • “Median of Medians” algorithm for finding exact median • Running time: O(n) • No faster algorithm possible for approximate median Can you give a short proof ? 5
  6. 6. ½ - Approximate median A Randomized Algorithm Rand-Approx-Median(A) 1. Let k  c log n; 2. S  ∅; 3. For i=1 to k 4. x  an element selected randomly uniformly from A; 5. S  S U {x}; 6. Sort S. 7. Report the median of S. Running time: O(log n loglog n) 6
  7. 7. Analyzing the error probability of Rand-approx-median n/4 Left Quarter Elements of A arranged in Increasing order of values 3n/4 Right Quarter When does the algorithm err ? To answer this question, try to characterize what will be a bad sample S ? 7
  8. 8. Analyzing the error probability of Rand-approx-median n/4 Elements of A arranged in Increasing order of values Left Quarter 3n/4 Median of S Right Quarter Observation: Algorithm makes an error only if k/2 or more elements sampled from the Right Quarter (or Left Quarter). 8
  9. 9. Analyzing the error probability of Rand-approx-median n/4 Elements of A arranged in Increasing order of values 3n/4 Right Quarter Left Quarter ¼ Exactly the same as the coin tossing exercise we did ! 9
  10. 10. Main result we discussed 10
  11. 11. ELEMENTARY PROBABILITY THEORY (IT IS SO SIMPLE THAT YOU UNDERESTIMATE ITS ELEGANCE AND POWER) 11
  12. 12. Elementary probability theory (Relevant for CS648) • We shall mainly deal with discrete probability theory in this course. • We shall take the set theoretic approach to explain probability theory. Consider any random experiment : o Tossing a coin 5 times. o Throwing a dice 2 times. o Selecting a number randomly uniformly from [1..n]. How to capture the following facts in the theory of probability ? 1. Outcome will always be from a specified set. 2. Likelihood of each possible outcome is non-negative. 3. We may be interested in a collection of outcomes. 12
  13. 13. Probability Space Ω 13
  14. 14. Event in a Probability Space A Ω 14
  15. 15. Exercises A randomized algorithm can also be viewed as a random experiment. 1. What is the sample space associated with Randomized Quick sort ? 2. What is the sample space associated with Rand-approx-median algorithm ? 15
  16. 16. An Important Advice In the following slides, we shall state well known equations (highlighted in yellow boxes) from probability theory. • You should internalize them fully. • We shall use them crucially in this course. • Make sincere attempts to solve exercises that follow. 16
  17. 17. Union of two Events A B Ω 17
  18. 18. Union of three Events A B C Ω 18
  19. 19. Exercises 19
  20. 20. Conditional Probability 20
  21. 21. Exercises • A man possesses five coins, two of which are double-headed, one is double-tailed, and two are normal. He shuts his eyes, picks a coin at random, and tosses it. What is the probability that the lower face of the coin is a head ? He opens his eyes and sees that the coin is showing heads; what it the probability that the lower face is a head ? He shuts his eyes again, and tosses the coin again. What is the probability that the lower face is a head ? He opens his eyes and sees that the coin is showing heads; what is the probability that the lower face is a head ? He discards this coin, picks another at random, and tosses it. What is the probability that it shows heads ? 21
  22. 22. Partition of sample space and an “important Equation” B Ω 22
  23. 23. Exercises 23
  24. 24. Independent Events P(A ∩ B) = P(A) · P(B) 24
  25. 25. Exercises 25

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