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Big O

Explore the divide and conquer technique in computer science, discussing problem-solving strategies, sorting algorithms, and runtime analysis. Understand the process of dividing problems, solving subproblems, and combining results for efficient solutions.

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Big O

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  1. Big O David Kauchak cs161 Summer 2009

  2. Administrative • Homework 1 is available and due Thur. at 5pm • Homework grading/policies • May turn in one homework late up to 2 days (48 hours) • For final grade, we’ll drop your lowest score • Note versioning • Final time: 8/14 3:30-6:30pm • Wed. office hour location change: Gates 195

  3. Divide and Conquer • Divide: Break the problem into smaller subproblems • Conquer: Solve the subproblems. Generally, this involves waiting for the problem to be small enough that it is trivial to solve (i.e. 1 or 2 items) • Combine: Given the results of the solved subproblems, combine them to generate a solution for the complete problem

  4. Divide and Conquer: some thoughts • Often, the sub-problem is the same as the original problem • Dividing the problem in half frequently does the job • May have to get creative about how the data is split • Splitting tends to generate run times with log n in them

  5. Divide and Conquer: Sorting • How should we split the data? • What are the subproblems we need to solve? • How do we combine the results from these subproblems?

  6. MergeSort

  7. MergeSort: Merge • Assuming L and R are sorted already, merge the two to create a single sorted array

  8. Merge L: 1 3 5 8 R: 2 4 6 7

  9. Merge L: 1 3 5 8 R: 2 4 6 7 B:

  10. Merge L: 1 3 5 8 R: 2 4 6 7 B:

  11. Merge L: 1 3 5 8 R: 2 4 6 7 B:

  12. Merge L: 1 3 5 8 R: 2 4 6 7 B: 1

  13. Merge L: 1 3 5 8 R: 2 4 6 7 B: 1

  14. Merge L: 1 3 5 8 R: 2 4 6 7 B: 1 2

  15. Merge L: 1 3 5 8 R: 2 4 6 7 B: 1 2

  16. Merge L: 1 3 5 8 R: 2 4 6 7 B: 1 2 3

  17. Merge L: 1 3 5 8 R: 2 4 6 7 B: 1 2 3

  18. Merge L: 1 3 5 8 R: 2 4 6 7 B: 1 2 3 4

  19. Merge L: 1 3 5 8 R: 2 4 6 7 B: 12 3 4

  20. Merge L: 1 3 5 8 R: 2 4 6 7 B: 12 3 4 5

  21. Merge L: 1 3 5 8 R: 2 4 6 7 B: 1 2 3 4 5

  22. Merge L: 1 3 5 8 R: 2 4 6 7 B: 1 2 3 4 5 6

  23. Merge L: 1 3 5 8 R: 2 4 6 7 B: 1 2 34 5 6

  24. Merge L: 1 3 5 8 R: 2 4 6 7 B: 1 2 34 5 6 7

  25. Merge L: 1 3 5 8 R: 2 4 6 7 B: 1 2 34 5 6 7

  26. Merge L: 1 3 5 8 R: 2 4 6 7 B: 1 2 34 5 6 7 8

  27. Merge • Does the algorithm terminate?

  28. Merge • Is it correct? • Loop invariant: At the end of each iteration of the for loop of lines 4-10 the subarray B[1..k] contains the smallest k elements from L and R in sorted order.

  29. Merge • Is it correct? • Loop invariant: At the end of each iteration of the for loop of lines 4-10 the subarray B[1..k] contains the smallest k elements from L and R in sorted order.

  30. Merge • Running time?

  31. Merge • Running time? Θ(n) - linear

  32. Merge-Sort: Divide 8 5 1 3 6 2 7 4 8 5 1 3 6 2 7 4

  33. Merge-Sort: Divide 8 5 1 3 6 2 7 4 8 5 1 3 6 2 7 4 8 5 1 3

  34. Merge-Sort: Divide 8 5 1 3 6 2 7 4 8 5 1 3 6 2 7 4 8 5 1 3 8 5

  35. Merge-Sort: Conquer 8 5 1 3 6 2 7 4 8 5 1 3 6 2 7 4 8 5 1 3 Sorting a list of one element is easy 8 5

  36. Merge-Sort: Combine 8 5 1 3 6 2 7 4 8 5 1 3 6 2 7 4 5 8 1 3 8 5

  37. Merge-Sort: Divide 8 5 1 3 6 2 7 4 8 5 1 3 6 2 7 4 5 8 1 3 1 3

  38. Merge-Sort: Divide 8 5 1 3 6 2 7 4 8 5 1 3 6 2 7 4 5 8 1 3 1 3

  39. Merge-Sort: Merge 8 5 1 3 6 2 7 4 8 5 1 3 6 2 7 4 5 8 1 3 1 3

  40. Merge-Sort: Merge 8 5 1 3 6 2 7 4 1 3 5 8 6 2 7 4 5 8 1 3

  41. Merge-Sort: … 8 5 1 3 6 2 7 4 1 3 5 8 6 2 7 4 …

  42. Merge-Sort: … 8 5 1 3 6 2 7 4 1 3 5 8 2 4 6 7

  43. Merge-Sort: Merge 1 2 3 4 5 6 7 8 1 3 5 8 2 4 6 7 Done!

  44. MergeSort • Running time? D(n): cost of splitting (dividing) the dataC(n): cost of merging/combining the data

  45. MergeSort • Running time? D(n): cost of splitting (dividing) the data - linear Θ(n)C(n): cost of merging/combining the data – linearΘ(n)

  46. MergeSort • Running time? Which is?

  47. MergeSort cn T(n/2) T(n/2)

  48. MergeSort cn cn/2 cn/2 T(n/4) T(n/4) T(n/4) T(n/4)

  49. MergeSort cn cn/2 cn/2 cn/4 cn/4 cn/4 cn/4 T(n/8) T(n/8) T(n/8) T(n/8) T(n/8) T(n/8) T(n/8) T(n/8)

  50. MergeSort cn cn/2 cn/2 cn/4 cn/4 cn/4 cn/4 cn/8 cn/8 cn/8 cn/8 cn/8 cn/8 cn/8 cn/8 … c c c c c … c c c c c c

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