# analysis techniques pasi fränti 19.9.2013. ordo o(g) – upper bound f(n) ≤ c∙g(n) omega ...

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• Analysis techniques19.9.2013

• Ordo O(g) Upper Bound f(n) cg(n)Omega (g) Lower Bound f(n) cg(n)Theta (g) Exact limit: c1g(n) f(n) c2g(n)

Upper and lower bounds

• Upper limitUpper limit

For example: f(n) = 3n-7 = O(n)Suppose that c= 4 then3n+7 4n 7 nwhich is true for all when n0=7

• Lower limitLower limit

Exact bounds

• Example of boundsT(N)=N-7T(N)=(N)N0=1N0=8T(N)=O(N)

Chart1

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T(N)=N-7

g1(N)=N

g2(N)=N/8

Sheet1

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110.13

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2013202.50

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Sheet1

T(N)=N-7

g1(N)=N

g2(N)=N/8

Sheet2

Sheet3

• Limit for polynomialsPolynomial function:Derivation of upper limit:

• LogarithmDefinition:Properties:

• Summations

• Combinatory formulasPermutation:Binomial factor:

• Lower bound for sorting

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

Depth of a binary treesN leaf nodes

• Initial case: For N=2, s = log2N=1Induction hypothesis: Trees of height s=k have at most N 2k leaves s log2NInduction 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

• 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 byProof of the lower bound

• Stirlings formula

More strict lower bound