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Unformatted text preview: Data Structures and Algorithm Analysis Instructor: Dr. Malek Mouhoub Computer Science Department University of Regina Fall 2010 CS340 Fall 2010 1 1. Algorithm Analysis 1. Algorithm Analysis • 1.1 Mathematics Review • 1.2 Introduction to Algorithm Analysis • 1.3 Asymptotic notation and Growth of functions • 1.4 Case Study • 1.5 Data Structures and Algorithm Analysis CS340 Fall 2010 2 1.1 Mathematics Review 1.1 Mathematics Review ¤ § ¥ ƒ Exponents X A X B = X A + B X A X B = X A B ( X A ) B = X AB X N + X N = 2 X N = X 2 N 2 N + 2 N = 2 N +1 CS340 Fall 2010 3 1.1 Mathematics Review ¤ § ¥ ƒ Logarithms • By default, logarithms used in this course are to the base 2. • X A = B ⇔ log X B = A • log A B = log C B log C A , A, B, C > , A = 1 • log AB = log A + log B ; A, B > • log A/B = log A log B • log ( A B ) = B log A • log X < X ∀ X > • log 1 = 0 , log 2 = 1 , log 1 , 024 = , log 1 , 048 , 576 = CS340 Fall 2010 4 1.1 Mathematics Review / £ ¡ ¢ Summations • ∑ N i =1 a i = a 1 + a 2 + ··· + a N • lim N →∞ ∑ N i =1 a i = ∑ ∞ i =1 a i = a 1 + a 2 + ··· (infinite sum) • Linearity – ∑ N i =1 ( ca i + db i ) = c ∑ N i =1 a i + d ∑ N i =1 b i – ∑ N i =1 Θ( f ( i )) = Θ( ∑ N i =1 f ( i )) • General algebraic manipulations : – ∑ N i =1 f ( N ) = N f ( N ) – ∑ N i = n f ( i ) = ∑ N i =1 f ( i ) ∑ n 1 i =1 f ( i ) CS340 Fall 2010 5 1.1 Mathematics Review / £ ¡ ¢ Summations • Geometric series : – ∑ N i =0 A i = A N +1 1 A 1 – if < A < 1 then ∑ N i =0 A i ≤ 1 1 A • Arithmetic series : – ∑ N i =1 i = N ( N +1) 2 = N 2 + N 2 ≈ N 2 2 – ∑ N i =1 i 2 = N ( N +1)(2 N +1) 6 ≈ N 3 3 – ∑ N i =1 i k ≈ N k +1  k +1  k = 1 * if k = 1 then H N = ∑ N i =1 1 i ≈ log e N * error in approx : γ ≈ , 57721566 CS340 Fall 2010 6 1.1 Mathematics Review / £ ¡ ¢ Products • Q N i =1 a i = a 1 × a 2 × ··· × a N • log( Q N i =1 a i ) = ∑ N i =1 log a i CS340 Fall 2010 7 1.1 Mathematics Review ¤ § ¥ ƒ Proving statements • Proof by induction 1. Proving a base case : establishing that a theorem is true for some small values. 2. Inductive hypothesis : the theorem is assumed to be true for all cases up to some limit k . 3. Given this assumption, show that the theorem is true for k + 1 • Proof by Counter example : find an example showing that the theorem is not true. • Proof by Contradiction : Assuming that the theorem is false and showing that this assumption implies that some known property is false, and hence the original assumption was erroneous. CS340 Fall 2010 8 1.2 Introduction to algorithm analysis 1.2 Introduction to algorithm analysis Boss gives the following problem to Mr Dupont, a fresh hired BSc in computer science (to test him . . . or may be just for fun) : T (1) = 3 T (2) = 10 T ( n ) = 2 T ( n 1) T ( n 2) What is T (100) ?...
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This note was uploaded on 11/15/2010 for the course CS 340 taught by Professor Dr.malek during the Fall '10 term at University of Regina.
 Fall '10
 Dr.Malek
 Data Structures

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