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CSE 780: Design and Analysis of Algorithms

CSE 780: Design and Analysis of Algorithms. Lecture 7: Prob. Analysis Randomized Alg. Expected Complexity. Probabilistic method: Given a distribution for all possible inputs Derive expected time based on distribution Randomized algorithm: Add randomness in the algorithm

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CSE 780: Design and Analysis of Algorithms

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  1. CSE 780: Design and Analysis of Algorithms Lecture 7: Prob. Analysis Randomized Alg. CSE 2331/5331

  2. Expected Complexity • Probabilistic method: • Given a distribution for all possible inputs • Derive expected time based on distribution • Randomized algorithm: • Add randomness in the algorithm • Analyze the expected behavior of the algorithm CSE 2331/5331

  3. First Example • What is worst case time complexity? • What is expected / average time complexity? CSE 2331/5331

  4. Expected Running Time • Expected / average running time • = probability of event I • = running time given event I • To analyze, need to assume a probabilistic distribution for all inputs CSE 2331/5331

  5. SeqSearchAlg • Expected running time = • If we assume • = 0 • All permutations are equally likely • implies • Then expected running time = CSE 2331/5331

  6. Remarks • For probabilistic analysis • An input probabilistic distribution input model has be assumed! • For a fixed input, the running time is fixed. • The average / expected time complexity is for if we consider running it for a range of inputs, what the average behavior is. • Randomized algorithm • No assumption in input distribution! • Randomness is added in the algorithm • For a fixed input, the running time is NOT fixed. • The expected time is what we can expect when we run the algorithm on ANY SINGLE input. CSE 2331/5331

  7. Randomized Algorithms CSE 2331/5331

  8. Expectation • X is a random variable • The expectation of X is • E.g, coin flip • Linearity of expectation: • Conditional expectation: CSE 2331/5331

  9. Linearity of Expectation • = expected running time for func2 • What is ? + cn CSE 2331/5331

  10. Conditional Expectation • = expected running time of func2 • = expected running time of func3 • What is the expected running time of func1? CSE 2331/5331

  11. A Randomized Example Worst case complexity? Expected case? CSE 2331/5331

  12. Running Time Analysis • Worst Case: • Expected running time: CSE 2331/5331

  13. CSE 2331/5331

  14. Time Analysis • Worst case: • Expected running time: CSE 2331/5331

  15. A more complicated variation. CSE 2331/5331

  16. Analysis • Worst case: • Expected case: CSE 2331/5331

  17. An Example with Recursion CSE 2331/5331

  18. Worst case: • Expected running time • Solving this we have CSE 2331/5331

  19. Another Example with Recursion CSE 2331/5331

  20. Analysis • Worst case: • ! • Expected running time: • ! CSE 2331/5331

  21. Another Example CSE 2331/5331

  22. Analysis • Worst case: • Expected running time: CSE 2331/5331

  23. Back To QuickSort • Randomized-QuickSort( A, 1, n ) • If ( )return; • s = Randomized-Partition ( A, 1, n ); • A1 = Randomized-QuickSort( A, 1, s-1 ); • A2 = Randomized-QuickSort( A, s+1, n ); CSE 2331/5331

  24. = expected running time of Randomized-QuickSort • Assume input array has no duplicates One can use substitution method to show that ET(n) = O(n lg n) CSE 2331/5331

  25. Another Way of Analysis CSE 2331/5331

  26. CSE 2331/5331

  27. Summary • Difference between probabilistic analysis and randomized algorithm analysis • Analyze a simple randomized algorithm • Linearity of expectation and conditional expectation • Randomized quicksort • expected running time CSE 2331/5331

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