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Divide and Conquer: Examples, Scenarios, and Recurrences

Learn about divide and conquer algorithms with examples and scenarios. Solve recurrences and analyze their running time in terms of Big-O notation.

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Divide and Conquer: Examples, Scenarios, and Recurrences

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  1. L18: Recurrences for Divide-and-Conquer Prob Prob 1 1 1 1 n/2 n-1 n n 1 1 n n L17 Subs Subs n/2 n 1 n/2+1 n-1 n-1 1 1 L18

  2. Page 2 Outline • D&C Examples • D&C Scenarios and Recurrences • Solving the recurrences • Big-O • Running time in terms of big-O • Three asymptotic behaviors of D&C recurrences • Reading: • Textbook: 228-240

  3. D&C Example: Merge Sort 7 2 4 3 1 8 6 5 Divide 1 8 6 5 7 2 4 3 Conquer 2 3 4 7 1 5 6 8 Merge 2 3 4 5 6 7 8 1

  4. D&C Example: Binary Search

  5. D&C Example: Binary Search

  6. Binary Recurrence

  7. Need ceiling function and deal with recurrences like the following What if n is not a power for 2?

  8. Different Types of Recurrences • Which case takes more time?

  9. Page 9 Outline • D&C Examples • D&C Scenarios and Recurrences • Solving the recurrences • Big-O • Running time in terms of big-O • Three Asymptotic behaviors of D&C recurrences

  10. D&C Scenarios and Recurrences

  11. Page 11 Outline • D&C Example : Binary search • D&C Scenarios and Recurrences • Solving the recurrences • Big-O • Running time in terms of big-O • Three Asymptotic behaviors of D&C recurrences

  12. Solving recurrence:

  13. Solving recurrence:

  14. Another Perspective: Recursion Tree

  15. Notes

  16. Notes

  17. Solving recurrence:

  18. Solving recurrence:

  19. Recursion Tree

  20. Page 20 Outline • D&C Example : Binary search • D&C Scenarios and Recurrences • Solving the recurrences • Big-O • Running time in terms of big-O • Three Asymptotic behaviors of D&C recurrences

  21. To get an clear appreciation of when running time differences that matters, we need big-O notation. How different are the running times?

  22. Asymptotically, 100n+10000 is better than

  23. Order of growth of f is upper bounded by order of growth ofg. Big-O/Asymptotic Upper Bound

  24. Big-O Example Show: 5n+10=O(n) Need to do: 6 10 Therefore, 5n+10=O(n) Proved.

  25. Big-O Examples g(n)=n2 In all cases, we can say the running time is O(n2). Convey essential info. & simple.

  26. Big-Theta

  27. Examples

  28. Use of big-O, ϴ If program has running time 3n2+n we can simply say: running time is ϴ(n2) Report dominating term only

  29. Page 30 Outline • D&C Example : Binary search • D&C Scenarios and Recurrences • Solving the recurrences • Big-O • Running time in terms of big-O • Three Asymptotic behaviors of D&C recurrences

  30. Running time in terms of Big-O

  31. Can further simplify using Instead, get big-O expression directly Solving recurrence: Assume n=2j

  32. Solving recurrence: Assume n=2j

  33. Running time in terms of Big-O

  34. Page 37 Outline • D&C Example : Binary search • D&C Scenarios and Recurrences • Solving the recurrences • Big-O • Running time in terms of big-O • Three Asymptotic behaviors of D&C recurrences

  35. Three Asymptotic Behaviors

  36. Case 2 proved already.

  37. Proof for Cases 1 & 3

  38. Case 1: a < 2

  39. Case 3: a > 2

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