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COP 3530 Spring2012

Data Structures & Algorithms

Discussion Session Week 5

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Outline

Growth of functions

Big Oh

Omega

Theta

Little Oh

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Growth of Functions

Gives a simple view of the algorithms efficiency.

Allows us to compare the relative performance of

alternative algorithms.f1(n) is O(n)

f2(n) is O(n^2)

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Growth of Functions

Exactrunning time of an algorithm is usually hardto

compute, and its unnecessary.

For large enough inputs, the lower-order terms ofan exact running time are dominated by high-order

terms.

f(n) = n^2 + 5n + 234n^2 >>5n + 234, when n is large enough

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Asymptotic Notation: Big Oh (O)

f(n)= O(g(n)) iff there exist positive constants c and n0 such thatf(n) cg(n) for all n n0

O-notation to give an upper bound on a function

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Asymptotic Notation: Big Oh (O)

Example 1[linear function] f(n) = 3n+2For n >= 2, 3n+2

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Asymptotic Notation: Big Oh (O)

Example 2[quadric function] f(n) = 10n^2+4n+2

For n >= 2, f(n) = 5, 5n < n^2.

Hence for n>= n0 = 5, f(n)

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Asymptotic Notation: Big Oh (O)

Example 3[exponential function] f(n) = 6*2^n + n^2For n>=4, n^2

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Asymptotic Notation: Big Oh (O)

Example 5[loose bounds] f(n) = 3n+3

For n >= 10, 3n+3

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Asymptotic Notation: Big Oh (O)

The specific values of cand n0used to satisfy the definition of bigoh are not important, we only say f(n) is big oh of g(n).

c/n0 do NOTmatter, WHY?

f(n)=n. Its close to 10n when comparing

with n^2, n^3

f(n) is relatively small when n

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Asymptotic Notation: Omega Notation

Big oh provides an asymptotic upperbound on a function.Omega provides an asymptotic lower bound on a function.

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Asymptotic Notation: Omega Notation

Example 7 f(n) = 3n+3 > 3n for all n. So f(n) = Omega(n)

Example 8[loose bounds] f(n) = 3n+3 > 1 for all n, so f(n) =

Omega(1)

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Asymptotic Notation: Theta Notation

Theta notation is used when functionfcan be bounded bothfrom above and belowby the same function g

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Asymptotic Notation: Theta Notation

Example 9: f(n) = 3n+3 is Theta(n), since n

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Asymptotic Notation: Little oh (o)

The asymptotic upper bound provided by O-notation may or maynotbe asymptotically tight. 2n = O(n) is tight, 2n = O(n^2) is not

tight.

We use o-notation to denote an upper bound that is NOT

asymptotically tight.

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Asymptotic Notation: Little oh (o)

f(n) = o(g(n)) ifff(n) = O(g(n)) and f(n) !=Omega(g(n))

Example 10 3n+2 = o(n^2) as 3n+2 = O(n^2) and 3n+2 !=

Omega(n^2)

Example 11 3n+2 != o(n) as 3n+2 = Omega(n)

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Review

Big oh: upperbound on a function.

Omega: lower bound.

Theta: lower and upper bound.- f(n) is Theta(g(n)) ifff(n) is O(g(n)) and Omega(g(n))

Little oh: loose upper bound.

- f(n) = o(g(n)) ifff(n) = O(g(n)) and f(n) !=Omega(g(n))

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Office Hour This Week:

Thursday 9thperiod at E309