Algorithms analysis and design

BYLecturer: Aisha Dawood

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stands alone on the right-hand side of an equation (or inequality), example : n = O(n2). means set membership :n O(n∈ 2).

when it appears in a formula, we interpret it as standing for some function that we do not care to name. For example, 2n2 + 3n + 1 = 2n2 + (n).

Left hand side and right hand side example : 2n2 + (n) = O(n2).

Order of growth notations in equations and inequalities

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the relational properties of real numbers apply to asymptotic comparisons as well.

Comparing functions

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For any two functions f(n) and g(n), we have f(n) = (g(n)), if and only if f(n) = O(g(n)) and f(n) = Ω(g(n)).

Proof: left as a home work….

Theorem

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Common expressions used to describe algorithm’s running time:

Name Expression

Constant 1 ,2,…,c

Logarithmic log(n)

Log squared log2(n)

Linear n

n log n n.log(n)

Quadratic n2

Cubic n3

Exponential 2n

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Conclusions (f(n)) describes f(n) f(n) = O(g(n)) and f(n) = Ω(g(n)). o(f(n) describes f(n) f(n) = O(g(n)) but not f(n) = Ω(g(n)). ω(f(n)) describes f(n) f(n) = Ω(g(n)) but not f(n) = O(g(n)).

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Rule #1: Sequential composition: Worst case running time of a sequence :

Rules For Big Oh Analysis of Running Time

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Rule #2: Iteration: Worst case running time of an iteration:

Rules For Big Oh Analysis of Running Time

For (S1)S2S3...SmO(T(n)) = MAX (O(T1(n)),O(T2(n)),...,O(Tm(n)))

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Rule #3: Conditional execution: Worst case running time of a conditional

statement :

Rules For Big Oh Analysis of Running Time

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Determine a tight big-oh bound on the running time of a program to compute the series of sums S0 , S1, S2..., Sn-1 where

T(n) = O(n2)

Example-Prefix Sums

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The Fibonacci numbers are the series of numbers F0,F1 ,F2 ... given by

Consider the sequence of Fibonacci numbers

T(n) = O(n)

Example-Fibonacci Numbers

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we use the definition of Fibonacci numbers to implement directly a recursive algorithm

Example-Fibonacci Numbers

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T(n) = Ω(Fn+1) By induction from the nature of the problem we get T(n) = Ω(2n)

Example-Fibonacci Numbers