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

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 a0,…,an−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 aiaj where
0≤i≠j≤n−1). Different elements here mean ai and aj with
i≠j (it can be the case that ai=aj).

5. Analysis and Design of Algorithms
Constraints 2≤n≤2 * 105 & 0≤a0,…,an−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:
𝐹𝑛 =
0 𝑛 = 0
1 𝑛 = 1
𝐹𝑛−2 + 𝐹𝑛−1 𝑛 > 1

27. Analysis and Design of Algorithms
Examples:
𝐹8 = 21
𝐹20 = 6765
𝐹50 = 12586269025
𝐹100 = 354224848179261915075

28. Analysis and Design of Algorithms
 Examples:
𝐹500 =
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 F6
 a=0, b=1
0 1
a b

42. Analysis and Design of Algorithms
 Advanced algorithm
 Compute F6
 a=b, b=a+b
1 1
a b

43. Analysis and Design of Algorithms
 Advanced algorithm
 Compute F6
 a=b, b=a+b
1 2
a b

44. Analysis and Design of Algorithms
 Advanced algorithm
 Compute F6
 a=b, b=a+b
2 3
a b

45. Analysis and Design of Algorithms
 Advanced algorithm
 Compute F6
 a=b, b=a+b
3 5
a b

46. Analysis and Design of Algorithms
 Advanced algorithm
 Compute F6
 a=b, b=a+b
5 8
a b

47. Analysis and Design of Algorithms
 Advanced algorithm
 Compute F6=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
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65. Analysis and Design of Algorithms
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