Analysis techniques Pasi Fränti 19.9.2013. Ordo O(g) – Upper Bound f(n) ≤ c∙g(n) Omega (g)– Lower Bound f(n) ≥ c∙g(n) Theta Θ(g)– Exact limit: c 1 ∙g(n)

Analysis techniques

Pasi Fränti19.9.2013

• Ordo O(g) – Upper Bound f(n) ≤ c∙g(n)• Omega (g) – Lower Bound f(n) ≥ c∙g(n)• Theta Θ(g) – Exact limit:

c1∙g(n) ≤ f(n) ≤ c2∙g(n)

Upper and lower bounds

3

• Upper limit

• For example: f(n) = 3n-7 = O(n)Suppose that c= 4 then

3n+7 ≤ 4n 7 ≤ n

which is true for all when n0=7

00 ),( :nc, if : )( nnngcnfgOf

Upper limit

4

• Lower limit

• Exact bounds

Lower limit

00 ),( :nc, if : )( nnngcnfgf

gfgfgf )(

Example of boundsT(N)=N-7

0

20

40

60

80

100

120

0 10 20 30 40 50 60 70 80 90 100

T (N )=N -7

g1(N )=N

g2(N )=N /8T(N)=(N)

N0=1

N0=8

T(N)=O(N)

Limit for polynomials

00

11

11)( nananananT m

mm

m

mmm

m

mmmm

mm

mm

mm

ncaaaaan

n

a

n

a

n

a

n

aan

nanananananT

)(

...

0121

01

12

21

00

11

22

11

Polynomial function:

Derivation of upper limit:

0121 selectingwhen aaaaac mm

mnOnT

Logarithm

)(log2 2 xyxy

)log(2

)log(1

2121

2121

2121

12

)log()log(

)log()log()log(

0)1log(

)log()log(

)log()log(

xx

a

xx

xax

xxxx

xxxx

xxxx

Definition:

Properties:

Summations

)(2

)1( 2

1

nnn

in

i

)(6

)12)(1( 3

1

2 nnnn

in

i

)(...21

1)1(

1

kkkn

i

k nn

k

ni

122 1

0

nn

i

i

1

11

0

a

aa

nn

i

i

Combinatory formulas

123...)2()1(! nnnn

Permutation:

)!(!

!

knk

n

k

n

Binomial factor:

Lower bound for sorting

a<b?

a<c? c<a?

c<b?b<c?

a b c

a c b c a b

c b a

b c a b a c

Sorting via comparisons

Theorem: The depth s of a binary tree with N leafs is s ≥ log2N

Depth of a binary tree

s

N leaf nodes

• Initial case: For N=2, s = log2N=1

• Induction hypothesis: Trees of height s=k have at most N ≤ 2k leaves s ≥ log2N

• Induction step: – Trees of height s=k+1 consists of one or two

sub-trees of size k. Denote the number of leaf nodes as N1 and N2 .

– According to induction hypothesis, N1 ≤ 2k and N2 ≤ 2k.

– Number of leafs N = N1+N2 ≤ 2k+2k = 2k+1. Hence, the hypothesis holds for k+1.

Proof of the height

k

K+1

k k

1

There are N! possible permutations, of which one is the list in sorted order. Tree has N! leaf nodes.

It can be lower bounded

Height of the tree is therefore bounded by

2/2/12...)2()1(! NNNNNN

)log(2/log!log 2/ NNNN N

Proof of the lower bound

nennn

nenn nn

n

nn

12

12!

12

12

6.0

Stirling’s formula

nnnn 22 log!log

More strict lower bound