Daa
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There are two main ways to show that an algorithm works correctly: testing and proofs of correctness. Testing an algorithm on sample inputs can find bugs but does not guarantee correctness. Proving correctness mathematically is better but more difficult. To prove the correctness of a recursive algorithm, it must be proved by mathematical induction that each recursive call solves a smaller subproblem and that the base cases are solved correctly. Examples are given of using induction to prove the correctness of recursive algorithms for calculating Fibonacci numbers, finding the maximum value, and multiplying numbers.
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Daa
- 1. Algorithm Correctness & Analysis
- 6. Recursive Fibonacci Numbers Claim: For all n ≥ 0 , fib( n ) return F n Base: For n =0 , fib( n ) returns 0 as claimed. For n = 1 , fib( n ) returns 1 as claimed. Induction: Suppose that n ≥ 2 and for all 0 ≤ m < n, fib ( m ) returns F m . Required to prove fib( n ) returns F n What does fib( n ) returns? fib( n-1 ) + fib( n-2 ) = F n-1 + F n-2 (by Ind. Hyp.) = F n
- 12. Analysis
- 13. Summary
- 15. Big oh
- 17. Note that we need Second example
- 18. Big Omega
- 19. Big Theta
- 20. True or false
- 21. Adding Big Ohs
- 22. Continue…
- 25. Example…
- 26. Time Complexity
- 27. Multiplication
- 28. Bubble sort
- 29. Analysis Trick
- 30. Example
- 31. Lies, Damn Lies, and Big-Os
- 33. Algorithms Analysis 2
- 34. The Heap
- 35. Contd…
- 36. But we have lost the tree structure To Delete the Minimum
- 37. But we have violated the structure condition Contd…
- 38. Contd…
- 39. Contd…
- 40. What Does it work?
- 41. Contd…
- 42. Contd…
- 43. To Insert a New Element
- 44. Contd…
- 45. Contd…
- 46. Contd…
- 47. Why Does it Work
- 48. Contd…
- 50. Analysis of Priority Queue Operation
- 51. Analysis of l (n)
- 52. Contd…
- 53. Example
- 54. Heapsort
- 55. Building a Heap Top Down
- 56. Contd…
- 57. Contd…
- 58. Building a Heap Bottom Up
- 59. Continue…
- 60. Continue…
- 63. Continue…
- 64. Example
- 65. Continue…
- 70. Warning
- 71. Reality Check
- 72. Merge Sorting
- 73. Continue…
- 74. Continue…
- 76. Proof Sketch
- 77. Continue…
- 79. Back to the Proof
- 80. Continue…
- 81. Messy Details
- 82. Continue…
- 83. Example