Algorithms Lecture 3: Analysis of Algorithms II

Algorithms Lecture 3: Analysis of Algorithms II

Benha University

http://www.bu.edu.eg

We will discuss the following: Maximum Pairwise Product, Fibonacci, Greatest Common Divisors, Naive algorithm is too slow. The Efficient algorithm is much better. Finding the correct algorithm requires knowing something interesting about the problem

4604d1f123000533847758729b23db6c?s=128

Mohamed Loey

October 30, 2017
Tweet

Transcript

  1. Analysis and Design of Algorithms Analysis of Algorithms II

  2. Analysis and Design of Algorithms Maximum Pairwise Product Fibonacci Greatest

    Common Divisors
  3. Analysis and Design of Algorithms Maximum Pairwise Product

  4. Analysis and Design of Algorithms Given a sequence of non-negative

    integers a 0 ,…,a n−1 , find the maximum pairwise product, that is, the largest integer that can be obtained by multiplying two different elements from the sequence (or, more formally, max a i a j where 0≤i≠j≤n−1). Different elements here mean a i and a j with i≠j (it can be the case that a i =a j ).
  5. Analysis and Design of Algorithms Constraints 2≤n≤2 * 105 &

    0≤a 0 ,…,a n−1 ≤105.
  6. Analysis and Design of Algorithms  Sample 1  Input:

    1 2 3  Output:6  Sample 2  Input: 7 5 14 2 8 8 10 1 2 3  Output:140
  7. Analysis and Design of Algorithms  Sample 3  Input:

    4 6 2 6 1  Output:36
  8. Analysis and Design of Algorithms  Assume the following array

    2 4 3 5 1
  9. Analysis and Design of Algorithms  Assume the following array

    2 4 3 5 1 i j Result=0
  10. Analysis and Design of Algorithms  Assume the following array

    2 4 3 5 1 i j If a[i]*a[j] > result result=a[i]*a[j]=8
  11. Analysis and Design of Algorithms  Assume the following array

    2 4 3 5 1 i j If a[i]*a[j] > result result=8
  12. Analysis and Design of Algorithms  Assume the following array

    2 4 3 5 1 i j If a[i]*a[j] > result result= a[i]*a[j] =10
  13. Analysis and Design of Algorithms  Naive algorithm

  14. Analysis and Design of Algorithms  Python Code:

  15. Analysis and Design of Algorithms  Time complexity O(n2)

  16. Analysis and Design of Algorithms we need a faster algorithm.

    This is because our program performs about n2 steps on a sequence of length n. For the maximal possible value n=200,000 = 2*105, the number of steps is 40,000,000,000 = 4*1010. This is too much. Recall that modern machines can perform roughly 109 basic operations per second
  17. Analysis and Design of Algorithms  Assume the following array

    2 4 3 5 1
  18. Analysis and Design of Algorithms  Find maximum number1 2

    4 3 5 1 max1
  19. Analysis and Design of Algorithms  Find maximum number2 but

    not maximum number1 2 4 3 5 1 max2 max1
  20. Analysis and Design of Algorithms  Find maximum number2 but

    not maximum number1 2 4 3 5 1 max2 max1 return max1*max2
  21. Analysis and Design of Algorithms  Efficient algorithm

  22. Analysis and Design of Algorithms  Efficient algorithm

  23. Analysis and Design of Algorithms  Time complexity O(n)

  24. Analysis and Design of Algorithms Fibonacci

  25. Analysis and Design of Algorithms 0,1,1,2,3,5,8,13,21,34, …

  26. Analysis and Design of Algorithms Definition:  = 0 =

    0 1 = 1 −2 + −1 > 1
  27. Analysis and Design of Algorithms Examples: 8 = 21 20

    = 6765 50 = 12586269025 100 = 354224848179261915075
  28. Analysis and Design of Algorithms  Examples: 500 = 1394232245616978801397243828

    7040728395007025658769730726 4108962948325571622863290691 557658876222521294125
  29. Analysis and Design of Algorithms  Naive algorithm

  30. Analysis and Design of Algorithms  Naive algorithm Very Long

    Time why????
  31. Analysis and Design of Algorithms

  32. Analysis and Design of Algorithms

  33. Analysis and Design of Algorithms F6 F5 F4 F3 F2

    F1 F0 F0 F1 F0 F2 F1 F0 F0 F3 F2 F1 F0 F0 F1 F0 F4 F3 F2 F1 F0 F0 F1 F0 F2 F1 F0 F0
  34. Analysis and Design of Algorithms Fib algorithm is very slow

    because of recursion Time complexity = O(2n)
  35. Analysis and Design of Algorithms  Efficient algorithm 0 1

    1 2 3 5 Create array then insert fibonacci
  36. Analysis and Design of Algorithms  Efficient algorithm

  37. Analysis and Design of Algorithms  Efficient algorithm short Time

    why????
  38. Analysis and Design of Algorithms Fib_Fast algorithm is fast because

    of loop + array Time complexity = O(n2)
  39. Analysis and Design of Algorithms  Efficient algorithm  Try

    Very long Time why????
  40. Analysis and Design of Algorithms Advanced algorithm No array Need

    two variable + Loop
  41. Analysis and Design of Algorithms  Advanced algorithm  Compute

    F 6  a=0, b=1 0 1 a b
  42. Analysis and Design of Algorithms  Advanced algorithm  Compute

    F 6  a=b, b=a+b 1 1 a b
  43. Analysis and Design of Algorithms  Advanced algorithm  Compute

    F 6  a=b, b=a+b 1 2 a b
  44. Analysis and Design of Algorithms  Advanced algorithm  Compute

    F 6  a=b, b=a+b 2 3 a b
  45. Analysis and Design of Algorithms  Advanced algorithm  Compute

    F 6  a=b, b=a+b 3 5 a b
  46. Analysis and Design of Algorithms  Advanced algorithm  Compute

    F 6  a=b, b=a+b 5 8 a b
  47. Analysis and Design of Algorithms  Advanced algorithm  Compute

    F 6 =8 5 8 a b
  48. Analysis and Design of Algorithms  Advanced algorithm

  49. Analysis and Design of Algorithms  Advanced algorithm Very short

    Time why????
  50. Analysis and Design of Algorithms Fib_Faster algorithm is faster because

    of loop + two variables Time complexity = O(n)
  51. Analysis and Design of Algorithms  Advanced algorithm  Try

    Short Time why????
  52. Analysis and Design of Algorithms Greatest Common Divisors

  53. Analysis and Design of Algorithms In mathematics, the greatest common

    divisor (gcd) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers.
  54. Analysis and Design of Algorithms Input Integers a,b >=0 Output

    gcd(a,b)
  55. Analysis and Design of Algorithms  What is the greatest

    common divisor of 54 and 24?  The divisors of 54 are: 1,2,3,6,9,18,27,54  Similarly, the divisors of 24 are: 1,2,3,4,6,8,12,24  The numbers that these two lists share in common are the common divisors of 54 and 24: 1,2,3,6  The greatest of these is 6. That is, the greatest common divisor of 54 and 24. gcd(54,24)=6
  56. Analysis and Design of Algorithms  Naive algorithm

  57. Analysis and Design of Algorithms

  58. Analysis and Design of Algorithms gcd algorithm is slow because

    of loop Time complexity = O(n) n depend on a,b
  59. Analysis and Design of Algorithms  Efficient algorithm

  60. Analysis and Design of Algorithms  Efficient algorithm

  61. Analysis and Design of Algorithms  Efficient algorithm gcd_fast((3918848, 1653264))

    gcd_fast((1653264, 612320)) gcd_fast((612320, 428624)) gcd_fast((428624, 183696)) gcd_fast((183696, 61232)) return 61232
  62. Analysis and Design of Algorithms Efficient algorithm Take 5 steps

    to solve gcd_fast( ( 3918848, 1653264 ) ) Time complexity = O(log(n))  n depend on a,b
  63. Analysis and Design of Algorithms Naive algorithm is too slow.

    The Efficient algorithm is much better. Finding the correct algorithm requires knowing something interesting about the problem.
  64. Analysis and Design of Algorithms facebook.com/mloey mohamedloey@gmail.com twitter.com/mloey linkedin.com/in/mloey mloey@fci.bu.edu.eg

    mloey.github.io
  65. Analysis and Design of Algorithms www.YourCompany.com © 2020 Companyname PowerPoint

    Business Theme. All Rights Reserved. THANKS FOR YOUR TIME