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Karatsuba Algorithm and Asymptotic Notation

This warm-up exercise explores the asymptotic runtime of the Karatsuba algorithm using the Tree method and introduces the concept of guess and check, induction, and the Master Theorem.

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Karatsuba Algorithm and Asymptotic Notation

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  1. CS4102 Algorithms Fall 2018 Warm Up What is the asymptotic run time of MergeSort if its recurrence is

  2. Tree method • per level levels of recursion … … … … …

  3. Today’s Keywords • Karatsuba • Guess and check Method • Induction • Master Theorem

  4. CLRS Readings • Chapter 4

  5. Homework • Hw1 due 11pm Wednesday, Sept 12 • Written (use Latex!) • Asymptotic notation • Recurrences • Divide and conquer

  6. Karatsuba Algorithm Recursively compute: , , Return a b c d Pseudo-code x = Karatsuba(a,c) y = Karatsuba(a,d) z = Karatsuba(a+b,c+d)-x-y Return 10nx +10n/2z+y

  7. Karatsuba 3. Use asymptotic notation to simplify … … … … … … … … …

  8. Karatsuba 3. Use asymptotic notation to simplify Math, math, and more math…(on board, see lecture supplemental)

  9. Recurrence Solving Techniques Tree Guess/Check “Cookbook” Substitution ? (induction)

  10. Induction (review) Goal: holds Base case(s): holds Hypothesis: holds Inductive step:

  11. Guess and Check Intuition • To Prove: • Consider: • Goal: show s.t. • (definition of big-O) • Technique: Induction • Base cases: • show for a small number of cases • Hypothesis: • Inductivestep:

  12. Karatsuba Guess and Check (Loose) Goal: Base cases: … up to some small Hypothesis: Inductive step:

  13. Mergesort Guess and Check Goal: Base cases: … up to some small Hypothesis: Inductive step:

  14. Karatsuba Guess and Check Goal: Base cases: by inspection, holds for small (at home) Hypothesis: Inductive step:

  15. What if we leave out the ? Goal: Base cases: by inspection, holds for small (at home) Hypothesis: Inductive step: Induction failed! What we wanted: What we got:

  16. “Bad Mergesort” Guess and Check Goal: Base cases: … up to some small Hypothesis: Inductive step:

  17. Recurrence Solving Techniques Tree Guess/Check “Cookbook” Substitution ?

  18. Observation • Divide: time, • Conquer: recurse on small problems, size • Combine: time • Recurrence: • Many D&C recurrences are of form:

  19. General … … … … … … … … … … … …

  20. 3 Cases Case 1: Most work happens at the leaves Case 2: Work happens consistently throughout Case 3: Most work happens at top of tree

  21. Master Theorem • Case 1: if for some constant , then • Case 2: if , then • Case 3: if for some constant , and if for some constant and all sufficiently large , then

  22. Proof of Case 1 , Insert math here… Conclusion:

  23. Master Theorem Example 1 • Case 1: if for some constant , then • Case 2: if , then • Case 3: if for some constant , and if for some constant and all sufficiently large , then Case 2

  24. Tree method … … … … …

  25. Master Theorem Example 2 • Case 1: if for some constant , then • Case 2: if , then • Case 3: if for some constant , and if for some constant and all sufficiently large , then Case 1

  26. Tree method … … … … … … … … … … …

  27. Master Theorem Example 3 • Case 1: if for some constant , then • Case 2: if , then • Case 3: if for some constant , and if for some constant and all sufficiently large , then Case 1

  28. Karatsuba … … … … … … … … …

  29. Master Theorem Example 4 • Case 1: if for some constant , then • Case 2: if , then • Case 3: if for some constant , and if for some constant and all sufficiently large , then Case 3

  30. Tree method … … … … …

  31. Homework Help Algorithm • Algorithm: How to ask a question about homework (efficiently) • Check to see if your question is already on piazza • If it’s not on piazza, ask on piazza • Look for other questions you know the answer to, and provide answers to any that you see • TA office hours • Instructor office hours • Email, set up a meeting

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