Asymptotic Analysis.ppt

Kinds of analyses
Worst-case: (usually)
• T(n) = maximum time of algorithm on any input of
size n.
Average-case: (sometimes)
• T(n) = expected time of algorithm over all inputs of
size n.
• Need assumption of statistical distribution of inputs.
Best-case:
• Cheat with a slow algorithm that works fast on some
input.

Asymptotic-O-notation
• For a given function , we denote by the set
of functions
• We use O-notation to give an asymptotic upper bound of
a function, to within a constant factor.
• means that there existes some constant c
s.t. is always for large enough n.
)
(n
g ))
(
( n
g
O










0
0
all
for
)
(
)
(
0
s.t.
and
constants
positive
exist
there
:
)
(
))
(
(
n
n
n
cg
n
f
n
c
n
f
n
g
O
))
(
(
)
( n
g
O
n
f 
)
(n
cg

)
(n
f

Ω-Omega notation
• For a given function , we denote by the
set of functions
• We use Ω-notation to give an asymptotic lower bound on
a function, to within a constant factor.
• means that there exists some constant c s.t.
is always for large enough n.
)
(n
g ))
(
( n
g












0
0
all
for
)
(
)
(
0
s.t.
and
constants
positive
exist
there
:
)
(
))
(
(
n
n
n
f
n
cg
n
c
n
f
n
g
))
(
(
)
( n
g
n
f 

)
(n
f )
(n
cg


-Theta notation
• For a given function , we denote by the set
of functions
• A function belongs to the set if there exist
positive constants and such that it can be “sand-
wiched” between and or sufficienly large n.
• means that there exists some constant c1
and c2 s.t. for large enough n.
)
(n
g ))
(
( n
g













0
2
1
0
2
1
all
for
)
(
)
(
)
(
c
0
s.t.
and
,
,
constants
positive
exist
there
:
)
(
))
(
(
n
n
n
g
c
n
f
n
g
n
c
c
n
f
n
g
)
(n
f ))
(
( n
g

1
c 2
c
)
(
1 n
g
c )
(
2 n
g
c
Θ
))
(
(
)
( n
g
n
f 

)
(
)
(
)
( 2
1 n
g
c
n
f
n
g
c 


Asymptotic notation
Graphic examples of and .

 ,
, O

2
2
2
2
1 3
2
1
n
c
n
n
n
c 


2
1
3
2
1
c
n
c 


Example 1.
Show that
We must find c1 and c2 such that
Dividing bothsides by n2 yields
For
)
(
3
2
1
)
( 2
2
n
n
n
n
f 



)
(
3
2
1
,
7 2
2
0 n
n
n
n 




o-notation
• We use o-notation to denote an upper bound
that is not asymptotically tight.
• We formally define as the set
))
(
( n
g
o
















0
0
all
for
)
(
)
(
0
s.t.
0
constants
a
exist
there
0
constant
positive
any
for
:
)
(
))
(
(
n
n
n
cg
n
f
n
c
n
f
n
g
o
0
)
(
)
(
lim 

 n
g
n
f
n

ω-notation
• We use ω-notation to denote an upper bound
that is not asymptotically tight.
• We formally define as the set
))
(
( n
g

















0
0
all
for
)
(
)
(
0
s.t.
0
constants
a
exist
there
0
constant
positive
any
for
:
)
(
))
(
(
n
n
n
f
n
cg
n
c
n
f
n
g




 )
(
)
(
lim n
g
n
f
n

Standard notations and common functions
• Floors and ceilings
    1
1 




 x
x
x
x
x

• Logarithms:
For all real a>0, b>0, c>0, and n
b
a
a
a
n
a
b
a
ab
b
a
c
c
b
b
n
b
c
c
c
a
b
log
log
log
log
log
log
log
)
(
log
log






• Logarithms:
b
a
c
a
a
a
a
b
a
c
b
b
b
b
log
1
log
log
)
/
1
(
log
log
log





• Factorials
For the Stirling approximation:
























n
e
n
n
n
n
1
1
2
!
0

n
)
lg
(
)
!
lg(
)
2
(
!
)
(
!
n
n
n
n
n
o
n
n
n






Asymptotic Analysis.ppt

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