CSCI2100C-Lecture4-Growth-BigOmegaBigTheta-Slides(1).pdf -...

CSCI2100C Data Structures
Complexity Analysis: Growth of
Functions
1
Sibo Wang
Department of Systems Engineering and Engineering Management
Chinese University of Hong Kong
Complexity Analysis: Growth of Functions

CSCI2100C Data Structures
Tutorial/Lab session
Every Thursday, 17:30 - 18:15 (from Week
3), @ERB 602
Week 3: Lab 1 session
Announcement
Complexity Analysis: Growth of Functions 2

CSCI2100C Data Structures
? 𝑛 = 𝑂(? 𝑛 )
if and only if we can find a positive constant ? and
𝑛
0
such that ? 𝑛 ≤ ? ⋅ ?(𝑛) for all 𝑛 ≥ 𝑛
0
.
Recap of Big-Oh Definition
Complexity Analysis: Growth of Functions 3
? ⋅ ?(𝑛)
?(𝑛)
𝑛
0
𝑛
Implication
? ⋅ ?(𝑛) grows faster than ?(𝑛)
Since it grows faster, it will
surpass ?(𝑛) at some point, and
we denote this point as 𝑛
0

CSCI2100C Data Structures
Rule ( i): The polynomial rule: The biggest eats all
? 𝑛 = 𝑛
3
+ 𝑛
2
+ 𝑛 .
𝑛
3
has the biggest power, it eats all other terms
• Therefore ? 𝑛 = 𝑂(𝑛
3
)
Rule (ii): Product property: The big multiple the big
? 𝑛 = 𝑛 + 1 ⋅ 𝑛 + 𝑛
5
.
The first big: 𝑂 𝑛 , The second big: 𝑂(𝑛
5
)
? 𝑛 = 𝑂 𝑛 ⋅ 𝑛
5
= 𝑂(𝑛
6
)
Rule (iii): Sum property: The bigger of the two big
? 𝑛 = 𝑛 + 1 + 𝑛 + 𝑛
5
.
The first big 𝑂(𝑛) , the second big: 𝑂 𝑛
5
? 𝑛 = 𝑂 max(𝑛, 𝑛
5
) = 𝑂 𝑛
5
Recap: Big-Oh Rules (i)-(v)
Complexity Analysis: Growth of Functions 4

CSCI2100C Data Structures
Rule (iv): log only beats constant
? 𝑛 = 𝑛
0.000001
+ log 𝑛
1000
= 𝑂(𝑛
0.000001
)
? 𝑛 = 0.00000001 ⋅ log 𝑛 + 100000 = 𝑂(log 𝑛)
Rule (v): exponential beats powers
? 𝑛 = 1.00001
?
+ 𝑛
999999
= 𝑂(1.00001
?
)
? 𝑛 = 3
?
+ 4
?
= 𝑂 4
?
Recap: Big-Oh Rules (i)-(v) (cont.)
Complexity Analysis: Growth of Functions 5

CSCI2100C Data Structures
Prove or disprove: ? 𝑛 = 2𝑛 = 𝑂(𝑛
2
)
Wait a second? Isn’t ? 𝑛 = 2𝑛 = 𝑂(𝑛) ?
Recall Big -Oh denotes the upper bound of the grow
rate of ?(𝑛) .
There might exist many upper bound
• ? 𝑛 = 2𝑛 = 𝑂(𝑛 ⋅ log 𝑛) = 𝑂 𝑛
2
= 𝑂 𝑛
3
= 𝑂(2
?
)
But we want to find the tightest.
• We also need the lower bound of the growth rate
Practice
Complexity Analysis: Growth of Functions 6
Fina a constant ?, 𝑛
0
and such that
2𝑛 ≤ ?𝑛
2
for all 𝑛 ≥ 𝑛
0
⇔ 2 ≤ ?𝑛 for all 𝑛 ≥ 𝑛
0
.