1. Big-O & Asymptotic Analysis

2. Big-O
Big – O : Implementation independent description of algorithm run time
O for Omicron
Technical: Function that by a constant factor, is an upper bound for steps in particular algorithm

3. Big-O
Big – O Practical def: Function that best describes how work grows in relation to problem size.
Example: n is a better match for the growth of 4n - 1 than n2

4. Functions

5. Big-O
Big – O Practical def: Function that best describes how work grows in relation to problem size.

6. Other bounds Big Omega : Big Theta :
function that is a lower bound by a constant factor
Big Theta :
function that can be either an upper bound or lower bound with right constant
Often mean Big-Theta when say Big-O

7. Big-O Mathematical Definition
Big – O : Function that by a constant factor, is an upper bound for steps in particular algorithm
𝑓 𝑛 𝑖𝑠 𝑂 𝑔 𝑛 if and only if There 𝑒𝑥𝑖𝑠𝑡 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑠 𝑐 𝑎𝑛𝑑 𝑘 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡
𝑓 𝑛 ≤𝑐∙𝑔 𝑛 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛 ≥𝑘
f(n) is work done by your code
g(n) is the function category you want to call it

8. Algebra of Big-O Show: 4n − 1 is O(n) Must pick c and k so that:
Pick k = 0 and c = 5
𝑓 𝑛 ≤𝑐∙𝑔 𝑛 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛 ≥𝑘 or
4n − 1 ≤𝑐∙𝑛 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛 ≥𝑘
4𝑛−1≤5𝑛 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛 ≥0

9. Algebra of Big-O
4n − 1 is O(n)
4𝑛−1≤5𝑛 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛 ≥0

10. Important Idea 1 Constants don’t matter in Big-O Expression Category
Can treat all constants as 1
Expression
Category 5n
n : Linear
3n2
n2 : Quadratic
n2/20
log2(n) + 6
log2(n) : Logarithmic 10
1 : Constant

11. Dominant Term Largest term drives growth curve n n2 + 20n n2 1 21 10
300
100
12,000
10,000
1000
1,020,000
1,000,000
10000
100,200,000
100,000,000

12. Example Show: 2n2 + 100n is O(n2) Must pick c and k so that:
Pick k = 100 and c = 5
𝑓 𝑛 ≤𝑐∙𝑔 𝑛 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛 ≥𝑘 or
2n n ≤𝑐∙𝑛2 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛 ≥𝑘
2n n≤5𝑛2𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛 ≥100

13. Example
2n n is O(n2)
2n n≤5𝑛2𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛 ≥100

14. Important Idea 2 For Big-O just report largest term
Examples: 8n2 + 3n  O(8n2)  O(n2) 3log2n + 4n  O(4n)  O(n) n  O(3n)  O(n)

15. BigO Limitations
Asymptotic Analysis : focus on behavior for large values

16. BigO Limitations BigO focuses on categories at scale
Results may not hold for small problem sizes

17. BigO Limitations BigO focuses on categories at scale
Other terms and constants may matter for algorithms in the same category

18. Cases Algorithm may have different Big-O for
Best case
Worst case
Average case
When in doubt, we describe the worst case