1. Big-O Analysis Example

2. Big O & Functions Calculating work: Normal line of code : 1
Loop : repetitions * work inside loop
Can apply BigO simplification to both parts
Function : work done inside function
Can apply BigO simplification

3. Big O Describes Categories
3 units of work = O(1)
int x = 2; //1
int y = x + 10; //1
double z = 1.5 * x + y * sqrt(y); //1

4. Big O Describes Categories
n * 1 unit of work = O(n)
int x = 2; //1
int y = x + 10; //1
double z = 1.5 * x + y * sqrt(y); //1

5. Example
Work for this function:
f(n) = (n-1) = 4n – 1 = O(n)

6. Example Simplify as we go… Inside loop = 4 = O(1)
N-1 loops that do 1 work= O(n * 1) = O(n)
+3 = O(n + 3) = O(n)

7. Loops Loops driving factor of Big-O
Pattern of loop counter tells repetitions
Loop
Pattern
Repetitions
for(int i = 0; i < n; i++)
… n – 1 n
for(int i = 0; i <= n; i++)
… n – 1 n
n + 1 = n
for(int i = 0; i < n; i += 2)
… n – 1
n / 2 = n
for(int i = 1; i < 2*n; i++)
… 2n
2n = n
for(int i = 1; i < n; i *= 2) …
logn

8. Logs Log patterns Loop counter multiplied
… for(int i = 1; i < n; i = i * 2)
Remaining work divided at each step
… counter = while(counter > 0) { counter /= 2; }

9. Log Fun Logs of differing base differ only by constant factor:
Base often unspecified in Big O
Generally 2 n
log2(n)
log10(n) 10
3.322 1
100
6.644 2
1000
9.966 3
10000
13.288 4

10. Nested Loops Nested loops get multiplied: n2 nlogn Loop Pattern
Repetitions
for(int i = 0; i < n; i++)
for(int j = 0; j < n; j++)
i = j = … n – 1 i = j = … n – 1
… i = n j = … n – 1
n * n n2
for(int i = 1; i <= n; i++) for(int j = 1; j <= n; j = j * 2)
i = j = … n i = j = … n i = j = … n i = j = … n … i = n j = … n
n * log2n
nlogn

11. Tricky Nested Loop What is BigO for these loops? Loop Pattern Work
for(int i = 0; i < n; i++) for(int j = 0; j <= i; j++)
i = j = 0 i = j = 0 1 i = j = i = j = … i = n – 1 j = … n – 1
1 2 3 4 … n

12. Tricky Nested Loop Sum of work 1+2+3+…+ 𝑛−3 + 𝑛−2 + (𝑛−1)
1+2+3+…+ 𝑛−3 + 𝑛−2 + 𝑛−1

13. ( 𝑛−1 2 𝑔𝑟𝑜𝑢𝑝𝑠)(𝑛 𝑝𝑒𝑟 𝑔𝑟𝑜𝑢𝑝)= 𝑛−1 2 𝑛 = 𝑛 2 −𝑛 2
Increasing Work
Sum of work:
1+2+3+…+ 𝑛−3 + 𝑛−2 + (𝑛−1)
1+2+3+…+ 𝑛−3 + 𝑛−2 + 𝑛−1
( 𝑛−1 2 𝑔𝑟𝑜𝑢𝑝𝑠)(𝑛 𝑝𝑒𝑟 𝑔𝑟𝑜𝑢𝑝)= 𝑛−1 2 𝑛 = 𝑛 2 −𝑛 2
𝑛 2

14. Tricky Nested Loop What is BigO for these loops? n2 Loop Pattern
Repetitions
for(int i = 0; i < n; i++)
for(int j = 0; j < n; j++)
i = j = … n – 1 i = j = … n – 1 i = j = … n – 1
… i = n j = … n – 1
n * n n2
for(int i = 0; i < n; i++) for(int j = 0; j <= i; j++)
i = j = 0 i = j = 0 1 i = j = i = j = … i = n – 1 j = … n – 1
n2 / 2

15. Full Analysis
What is Big-O?

16. Big O & Functions
What is n?

17. Big O & Functions
N = size of array

18. Big O & Functions Declaring variable = constant
Initializing n values = O(n)

19. Print printArray = Loop + O(1)
Loop repeats n times, has 1 unit of work inside

20. Print
printArray = Loop + O(1) = O(n * 1) = O(n)

21. So Far
So far: 3n + 1 = O(N)

22. Sort
Loops N time

23. Sort
Assignments are all O(1)

24. Sort
Total work in loop = 4* O(1) + find Location

25. Find
findLocation = do constant work N times
O(n)

26. Sort
Total work in loop
= 4*O(1) + O(n)
= O(1) + O(n)
= O(n)

27. Sort
While loop repeats n times
Does O(n) work
Sort
= O(n * n) = O(n2)

28. Main
3N N2 = O(N2)
Total work of program O(n2)