이산수학 (Discrete Mathematics)
2.2 함수의 증가 (Growth of Functions)
2006년 봄학기
문양세
강원대학교 컴퓨터과학과
Orders of Growth – Introduction
2.2 Growth of Functions
For functions over numbers, we often need to know a
rough measure of how fast a function grows.
(과연 함수는 얼마나 빨리 증가하는가?)
If f(x) is faster growing than g(x), then f(x) always
eventually becomes larger than g(x) in the limit (for large
enough values of x).
(f(x)가 g(x)보다 빨리 증가한다면, 궁극적으로 f(x)의 값이 g(x)의 값보다 커진다.)
Useful in engineering for showing that one design scales
better or worse than another.
(공학에서는 어느 것이 다른 것보다 얼마나 나쁜지/좋은지를 보여주는 척도(의
정도)가 필요하다…)
Page 2
Discrete Mathematics
by Yang-Sae Moon
Orders of Growth – Motivation
2.2 Growth of Functions
Suppose you are designing a web site to process user data.
Suppose database
• program A takes fA(n)=30n+8 microseconds to process any n records,
while
• program B takes fB(n)=n2+1 microseconds to process the n records.
Which program do you choose, knowing you’ll want to
support millions of users?
Page 3
Discrete Mathematics
by Yang-Sae Moon
On a graph, as
you go to the
right, a faster
growing function
eventually
becomes larger...
Value of function 
Visualizing Orders of Growth
2.2 Growth of Functions
fA(n)=30n+8
fB(n)=n2+1
Increasing n 
Page 4
Discrete Mathematics
by Yang-Sae Moon
Concept of Orders of Growth
2.2 Growth of Functions
We say fA(n)=30n+8 is order n, or O(n). It is, at most,
roughly proportional to n.
fB(n)=n2+1 is order n2, or O(n2). It is roughly proportional
to n2.
Any O(n2) function is faster-growing than any O(n) function.
For large numbers of user records, the O(n2) function will
always take more time.
Page 5
Discrete Mathematics
by Yang-Sae Moon
Definition: O(g), at most order g
2.2 Growth of Functions
Let g be any function RR.
Define “at most order g”, written O(g), to be:
{f:RR | c,k: x>k: f(x)  cg(x)}.
• “Beyond some point k, function f is at most a constant c times g
(i.e., proportional to g).”
•
(x > k일 때, f(x)  cg(x)를 만족하는 c, k가 존재하면, f는 O(g)라 한다.)
“f is at most order g”, or “f is O(g)”, or “f=O(g)” all just
mean that fO(g).
Sometimes the phrase “at most” is omitted.
(일반적으로 그냥 “오더 g”라 이야기한다.)
Page 6
Discrete Mathematics
by Yang-Sae Moon
Points about the Definition
2.2 Growth of Functions
Note that f is O(g) so long as any values of c and k exist
that satisfy the definition.
But: The particular c, k, values that make the statement
true are not unique: Any larger value of c and/or k will
also work. (조건을 만족하는 c, k는 여러 개 있을 수 있다.)
You are not required to find the smallest c and k values
that work. (Indeed, in some cases, there may be no
smallest values!)
(조건을 만족하는 가장 작은 c, k를 찾을 필요는 없다. 보여주기만 하면 된다.)
However, you should prove that the values you choose do
work. (증명하기 위해서는 만족하는 c, k를 찾아야 한다.)
Page 7
Discrete Mathematics
by Yang-Sae Moon
“Big-O” Proof Example
2.2 Growth of Functions
Show that 30n+8 is O(n).
• Show c,k: n>k: 30n+8  cn.
• Let c=31, k=8. Assume n>k(=8).
• Then cn = 31n = 30n + n > 30n+8, so 30n+8 < cn.
Show that n2+1 is O(n2).
• Show c,k: n>k: n2+1  cn2.
• Let c=2, k=1. Assume n>1.
• Then cn2 = 2n2 = n2+n2 > n2+1, or n2+1< cn2.
Page 8
Discrete Mathematics
by Yang-Sae Moon
Note 30n+8 isn’t less
than n anywhere (n>0).
It isn’t even less than
31n everywhere.
But it is less than 31n
everywhere to the right
of n=8.
Value of function 
Big-O Example, Graphically
2.2 Growth of Functions
cn =
31n
30n+8
n
30n+8
O(n)
n>k=8 
Increasing n 
Page 9
Discrete Mathematics
by Yang-Sae Moon
Examples of Big-O (1/2)
2.2 Growth of Functions
f(x) = anxn + an-1xn-1 + … + a1x + a0
• f(x) = anxn + an-1xn-1 + … + a1x + a0
 |an|xn + |an-1|xn-1 + … + |a1|x + |a0|
= xn(|an| + |an-1|/x + … + |a1|/xn-1 + |a0|/xn)
 xn(|an| + |an-1| + … + |a1| + |a0|)
= cxn = O(xn)
c
f(n) = 1 + 2 + 3 + … + n
• f(n) = 1 + 2 + 3 + … + n
n+n+n+…+n
= n2 = O(n2)
Page 10
Discrete Mathematics
by Yang-Sae Moon
Examples of Big-O (2/2)
2.2 Growth of Functions
f(n) = n!
• f(n) = n! = n∙(n-1)∙(n-2)∙…∙3∙2∙1
 n∙n∙n∙…∙n∙n∙n
= nn = O(xn)
f(n) = log(n!)
• f(n) = log(n!)
 log(nn)
= nlogn = O(nlogn)
Page 11
Discrete Mathematics
by Yang-Sae Moon
Useful Facts about Big-O (1/2)
2.2 Growth of Functions
Big O, as a relation, is transitive:
fO(g)  gO(h)  fO(h)
Sums of functions:
If gO(f) and hO(f), then g+hO(f).
c>0, O(cf)=O(f+c)=O(fc)=O(f)
• E.g., O(2x2) = O(x2+2) = O(x2-3) = O(x2)
If f1=O(g1) and f2 = O(g2), then f1+f2 = O(max(g1,g2))
• E.g., if f1(x) = x2 = O(x2), f2(x) = x3 = O(x3),
then (f1+f2)(x) = x2 + x3 = O(x3) = O(max(x2, x3))
Page 12
Discrete Mathematics
by Yang-Sae Moon
Useful Facts about Big-O (2/2)
2.2 Growth of Functions
f1=O(g1) and f2 = O(g2), then f1f2 = O(g1g2)
• E.g., if f1(x) = x2 = O(x2), f2(x) = x3 = O(x3),
then (f1f2)(x) = x2∙x3 = x5 = O(x3) = O(x2∙x3))
Other useful facts…
•  f,g & constants a,bR, with b0,
- af = O(f)
- f+O(f) = O(f)
(e.g. 3x2 = O(x2))
(e.g. x2+x = O(x2))
• Also, if f=(1) (at least order 1), then:
-
|f|1-b = O(f);
(logb |f|)a = O(f)
g=O(fg)
fg  O(g)
a=O(f)
(e.g. x1 = O(x))
(e.g. log x = O(x))
(e.g. x = O(x log x))
(e.g. x log x  O(x))
(e.g. 3 = O(x))
Page 13
Discrete Mathematics
by Yang-Sae Moon
Definition: (g), exactly order g
2.2 Growth of Functions
If fO(g) and gO(f) then we say “g and f are of the same
order” or “f is (exactly) order g” and write f(g).
(f(x)=O(g(x))이고 g(x)=O(f(x))이면, f(x)=(g(x))이며 g(x)=(f(x))이다.)
Another equivalent definition:
(g)  {f:RR | c1c2k x>k: |c1g(x)||f(x)||c2g(x)| }
(c1g(x)  f(x)  c2g(x)를 만족하는 c1과 c2가 존재하면 f(x)=(g(x))이다.)
One more definition: (g) = {f | gO(f)}
(i.e., g(x)=O(f(x))
“The functions that are at least order g.” (하한 정의)
Page 14
Discrete Mathematics
by Yang-Sae Moon
Rules for 
2.2 Growth of Functions
Mostly like rules for O( ), except:
f,g>0 & constants a,bR,
af  (f), but
f  (fg) unless g=(1)
|f|1-b  (f), and
(logb|f|)c  (f).
with b>0,
 Same as with O.
 Unlike O.
 Unlike with O.
 Unlike with O.
Page 15
Discrete Mathematics
by Yang-Sae Moon
Examples of  (1/2)
2.2 Growth of Functions
f(n) = 1 + 2 + 3 + … + n = (n2)?
• f(n) = 1 + 2 + 3 + … + n
n+n+n+…+n
= n2
• f(n) = 1 + 2 + 3 + … + n
= (n∙(n – 1))/2
= n2/2 – n/2
 n2/2
• n2/2  f(n)  n2, i.e., c1 = ½, c2 = 1
 f(n) = (n2)
Page 16
Discrete Mathematics
by Yang-Sae Moon
Examples of  (2/2)
2.2 Growth of Functions
f(y) = 3y2 + 8ylogy = (y2)?
• f(n) = 3y2 + 8ylogy
 11y2 (if y > 1)
(since 8ylogy  8y2)
• f(n) = 3y2 + 8ylogy
 y2
(if y > 1)
• y2  f(y)  11y2, i.e., c1 = 1, c2 = 11
 f(y) = (y2)
Page 17
Discrete Mathematics
by Yang-Sae Moon