1. BY Lecturer: Aisha Dawood

2.  stands alone on the right-hand side of an equation (or inequality), example : n = O(n 2 ). 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, 2n 2 + 3n + 1 = 2n 2 +  ( n).  Left hand side and right hand side example : 2n 2 +  ( n) = O(n 2 ). 2

3.  the relational properties of real numbers apply to asymptotic comparisons as well. 3

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5.  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…. 5

6.  Common expressions used to describe algorithm’s running time: 6 NameExpression Constant1, 2,…,c Logarithmiclog(n) Log squaredlog 2 (n) Linearn n log nn.log(n) Quadraticn2n2 Cubicn3n3 Exponential2n2n

7. 7   (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)).

8.  Rule #1:  Sequential composition:  Worst case running time of a sequence : 8

9.  Rule #2:  Iteration:  Worst case running time of an iteration: 9 For (S1) S2 S3. Sm O(T(n)) = MAX (O(T1(n)),O(T2(n)),...,O(Tm(n))) For (S1) S2 S3. Sm O(T(n)) = MAX (O(T1(n)),O(T2(n)),...,O(Tm(n)))

10.  Rule #3:  Conditional execution:  Worst case running time of a conditional statement : 10

11.  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(n 2 ) 11

12.  The Fibonacci numbers are the series of numbers F0,F1,F2... given by  Consider the sequence of Fibonacci numbers  T(n) = O(n) 12

13.  we use the definition of Fibonacci numbers to implement directly a recursive algorithm 13

14.  T(n) = Ω(Fn+1)  By induction from the nature of the problem we get T(n) = Ω(2 n ) 14