1
20
Theta notation
Theta:
Often confused with
Big-Oh
notation !
Example:
Claim: Low order terms do not matter.
Needs a proof !
(HW?)
f n
g n
c c n
n
n
c g n
f n
c g n
( )
( ( ))
,
,
.
:
( )
( )
( )
const
s.t
1
2
1
2
0
0
0
2
2
0
0
2
2
2
2
2
2
2
2
2
2
1
2
/2
2
(
)
Proof:
take
8, then for
:
/2
2
/4
/4
2
/4
8 /4
2
/4
On the other hand, we have:
/2
2
/2
Thus:
/4
/2
2
/2
i.e.
1/4,
1/2.
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
c
c
Lecture 2, Sept. 23, 2010
21
Simple Theorem
Claim:
f(n) = O(g(n)) and g(n) = O(f(n))
⇒
f(n) =
(g(n))
1
1
1
1
2
2
2
2
1
2
1
2
Proof:
,
s.t.
n
n : 0
( )
( )
,
s.t.
n
n : 0
( )
( )
1
max(
,
): 0
( )
( )
( )
QED
n c
f n
c g n
n
c
g n
c f n
n
n n
g n
f n
c g n
c

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2
22
Summary
Remember the definitions.
Formally prove from definitions.
Use intuition from the properties of
“
”, “
“,
etc.
Consider behavior of f(n)/g(n) as n
→ ∞
Example of an algorithm
Stable Marriage
n
man and
n
women
Each woman ranks all men and each man ranks all women
Find a way to match (marry) all men and women such that
there are no two pairs
(m,w)
and
(m’,w’)
that are married
and such that
»
m
prefers
w’
to
w
»
w’
prefers
m
to
m’
In other words,
m
will “steal”
w’
from
m’
and
w’
will agree
Red line shows “instability”
23
m
m’
w
w’