2 then 1 2 1 2 example 3 2 1 7 12 what is the time

𝑛 = 𝑂(?
2
𝑛 )
• Then, ?
1
𝑛 ⋅ ?
2
𝑛 = 𝑂(?
1
𝑛 ⋅ ?
2
(𝑛))
Example
? 𝑛 = (𝑛
3
+ 𝑛
2
+ 1)(𝑛
7
+ 𝑛
12
)
What is the time complexity of ? 𝑛
• Let ?
1
𝑛 = (𝑛
3
+ 𝑛
2
+ 1) , ?
2
𝑛 = (𝑛
7
+ 𝑛
12
)
– ?
1
𝑛 = 𝑂 𝑛
3
, ?
2
𝑛 = 𝑛
12
– ? 𝑛 = ?
1
𝑛 ⋅ ?
2
𝑛 = 𝑂 𝑛
3
⋅ 𝑛
12
= 𝑂(𝑛
15
)
Product Rule
Complexity Analysis: Growth of Functions 14
The big multiple the big

CSCI2100C Data Structures
Sum property
?
1
𝑛 = 𝑂 ?
1
𝑛 , ?
2
𝑛 = 𝑂(?
2
𝑛 )
• Then, ?
1
𝑛 + ?
2
𝑛 = 𝑂 max(?
1
𝑛 , ?
2
𝑛 )
Example
? 𝑛 = 𝑛
3
+ 𝑛
2
+ 1 + (𝑛
4
+ 𝑛
7
+ 𝑛
20
+ 𝑛
30
)
?
1
= 𝑛
3
+ 𝑛
2
+ 1 , ?
2
= 𝑛
4
+ 𝑛
7
+ 𝑛
20
+ 𝑛
30
• ?
1
= 𝑂(𝑛
3
)
• ?
2
= 𝑂(𝑛
30
)
? 𝑛 = 𝑂 max(𝑛
3
, 𝑛
30
)
• 𝑂 𝑛
30
Big-Oh Rules (iii)
Complexity Analysis: Growth of Functions 15
The bigger of the two big

CSCI2100C Data Structures
We denote the number of basic operations of
each line as a function of n
?
1
𝑛 , ?
2
𝑛 , ⋯
LinearSearch: Using Sum Property
Complexity Analysis: Growth of Functions 16
LinearSearch(A, n, searchnum)
for i = 0 to n -1
if a [i] = searchnum
return i
return -1
1
2
3
4
2*n +2
2*n
1
1
Total: 4*n+4
?
1
𝑛 = 𝑂(𝑛)
?
2
𝑛 = 𝑂(𝑛)
?
3
𝑛 = 𝑂(1)
?
4
𝑛 = 𝑂(1)
𝑂 max(𝑛, 𝑛 , 1,1 ) = 𝑂(𝑛)

CSCI2100C Data Structures
What is wrong here?
.
Complexity Analysis: Growth of Functions 17

CSCI2100C Data Structures
Log functions grow slower than power functions.
lim
𝑛→∞
log 𝑛
?
𝑛
?
→ 0 , for ?, ? > 0
If ? 𝑛 = log 𝑛
?
+ 𝑛
?
where ?, ? > 0,
Then ?(𝑛) = 𝑂(𝑛
?
) .
If ? 𝑛 = log 𝑛
?
+ ? , where ?, ? > 0,
Then ?(𝑛) = 𝑂 log 𝑛
?
.
Example
? 𝑛 = log 𝑛
2
+ 𝑛
0.0001
• Let ? = 2, ? = 0.0001
• 𝑂(𝑛
0.0001
)
Big-Oh Rules (iv)
Complexity Analysis: Growth of Functions 18
log only beats constant

What is the time complexity of ?
2
)
What is the time complexity of