CS336f1011 - 10/6/10 Lecture 11 CS336 F10 Definition
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Unformatted text preview: 10/6/10 Lecture 11 CS336 F10 Definition
 Asymptotic Dominance Let f and g be functions from N to R. Then f is asymptotically dominated by g or f is O(g) if (∃k,C| : (∀x| x>k: |f(x)| ≤ C|g(x)|)) That is, there exists constants k and C such that |f(x)| ≤ C|g(x)| for all x>k Prove
that
f
is
O(f)
 Proof:
 |f(x)|
≤|f(x)|
for
all
x
 Hence
by
definition
of
big
O
with
C=1
and
k=0,
 we
conclude
f
is
O(f).
 Example

 •  Let
k
be
a
positive
integer.

Show
that
if

 
 f(n)=log(Πi|1≤i≤n:ik)
then
f(n)
is
O(nlogn).
 Example:

Find
C
and
k
such
that

 |log(Πi|1≤i≤n:ik)
| ≤ C|nlogn| for all n≥k.

 Proof:
 |log(Πi|1≤i≤n:ik)
| Example:

Find
C
and
k
such
that

 |log(Πi|1≤i≤n:ik)
| ≤ C|nlogn| for all n≥k.

 Proof:
 |log(Πi|1≤i≤n:ik)
| ≤ < log is ≥0 so ignore ||;since i≤n> log(Πi|1≤i≤n:nk)
 1 10/6/10 Example:

Find
C
and
k
such
that

 |log(Πi|1≤i≤n:ik)
| ≤ C|nlogn| for all n≥k.

 Proof:
 |log(Πi|1≤i≤n:ik)
| ≤ < log is ≥0 so ignore ||;since i≤n> log(Πi|1≤i≤n:nk)
 =<arith>
 log
(nk)n

for
n≥1
 Example:

Find
C
and
k
such
that

 |log(Πi|1≤i≤n:ik)
| ≤ C|nlogn| for all n≥k.

 Proof:
 |log(Πi|1≤i≤n:ik)
| ≤ < log is ≥0 so ignore ||;since i≤n> log(Πi|1≤i≤n:nk)
 =<arith>
 log
(nk)n

for
n≥1
 =<arith>
 kn(log
n)
for
n≥1
 Hence
by
the
definition
of
big‐O
with
C=k
and
k=1
 
 log(Πi|1≤i≤n:ik)

is
O(nlogn).
 Uses
for
Big
O 
 •  Common
estimates
 Definition
 Let f and g be functions from N to R. Then f asymptotically dominates g or f is Ω(g) (∃k,C| : (∀x| x>k: |f(x)| ≥ C|g(x)|)) That is, there exists constants k and C such that |f (x)| ≥ C|g(x)| for all x>k •  If
f
is
O(g)
then
g
is
Ω(f). •  If
f
is
O(g)
and
f
is
Ω(g) then f is Θ(g). For
each
of
the
following
statements,
 
 determine
if
they
are
true
or
false.
 
 •  •  •  •  •  •  •  xlogx
is
O(x2).

 
 
 
 True
 
 False

 xlogx
is
Ω(x2).
 
 
 
 True
 
 False

 xlogx
is
Θ(x2).
 
 
 
 True
 
 False

 2n
is
O(n!)

 
 
 
 
 True
 
 False

 If
f1
is
O(g)
and
f2
is
O(g)
then
f1
‐

f2
is
O(g)

 True
 False
 If
f1
is
Θ(g)
and
f2
is
Θ
(g)
then
f1‐f2
is
Θ
(g)

 True
 False

 If
f
is
Θ(g)
then
f
is
O(g)

 
 True
 
 False

 Quiz
13
 
 For
each
of
the
following
statements,
 determine
if
they
are
true
or
false
and
circle
 appropriately.

 a.

(5x2+x)(2logx‐1)
is
Ω
(x2).
 True
 
 False
 

 b.

logx3
is
O(x2).

 
 
 
 
 True
 
 False
 

 c.  If
f1
is
Θ(g1)
and
f2
is
Θ(g2),
then
f1‐f2
is

 
 Θ(g1‐g2).
 
 True
 
 False
 2 10/6/10 Example

 Example

 •  Prove
or
disprove
that
if
0<a<b,
the
na
is
O(nb)
 3 ...
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