Zeinab EidAlgorithm Analysis1 Chapter 4 Analysis Tools

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Zeinab EidAlgorithm Analysis3 Why we Analyze Algorithms We try to save computer resources such as:  Memory space used to store data structures, Space Complexity  CPU time, which is reflected by Algorithm Run Time Complexity.

Zeinab EidAlgorithm Analysis4 4.1 Seven Functions 1.The Constant Function: f(n)= c, where c is some fixed constant, c=5, c=100, or c=2 10, so we let g(n)=1, so f(n)=cg(n) 2.The Logarith Function: f(n)= log b n x = log b n iff b x =n, (see Proposition 4.1), the most common base for logarithmic function in computer science is 2. 3.Linear Function: f(n)= n, important for algorithms on vectors (one dim. Arrays).

Zeinab EidAlgorithm Analysis5 4.The N-Log-N Function: f(n)= nlogn This function grows a little faster than linear function and a lot slower than quadratic function. 5.The Quadratic Function: f(n)= n 2 It’s important for algorithms of nested loops. 6.The Cubic Function (Polynomial): f(n)= n 3 f(n)=a 0 +a 1 n+ a 2 n 2 +…+a d n d, is a polynomial of degree d, where a 0, a 1,…., a d are constants. 7.The Exponential Function: f(n)= b n

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Zeinab EidAlgorithm Analysis Primitive Operations If we wish to analyze

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Zeinab EidAlgorithm Analysis15 We define a set of primitive operations such as the following: Assigning a value to a variable Calling a method Performing an arithmetic operation (for example, adding two numbers) Comparing two numbers Indexing into an array Following an object reference Returning from a method.

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Zeinab EidAlgorithm Analysis18 Primitive Operations

Zeinab EidAlgorithm Analysis19 Counting Primitive Operations

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Zeinab EidAlgorithm Analysis Asymptotic Notation

Zeinab EidAlgorithm Analysis24 Example

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Zeinab EidAlgorithm Analysis27 Properties of Big-Oh Notation

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Zeinab EidAlgorithm Analysis34 Big-Omega and Big-Theta

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Zeinab EidAlgorithm Analysis36 Big-Omega Just as Big-Oh notation provides us a way of saying that a function f(n) is “less than or equal to” another function, Big-Omega notation provides us a way of saying that a function f(n) is “greater than or equal to” another function. Let f(n) and g(n) be functions mapping non- negative integers: We say that f(n) is Ω(g(n)) if g(n) is O(f(n)) That’s there is a real constant c>0 and an integer constant n 0 ≥ 1 such that: f(n) ≥ cg(n), for n ≥ n 0

Zeinab EidAlgorithm Analysis37 Big-Theta In addition, there is a notation that allows us to say that two functions grow at the same rate, up to a constant factor. We say that f(n) is Θ(g(n)) if f(n) is O(g(n)) and f(n) is Ω(g(n)), That’s, there are real constants c’>0 and c”>0 and an integer constant n 0 ≥ 1 such that: c’g(n) ≤ f(n) ≥ c”g(n), for n ≥ n 0.

Zeinab EidAlgorithm Analysis38 Example f(n) = 3nlog n + 4n + 5log n is Θ(nlog n) Since 3nlog n + 4n + 5log n ≥ 3nlog n for n ≥2 So, f(n) is Ω(nlog n) Since 3nlog n + 4n + 5log n ≤ (3+4+5)nlog n for n ≥2 So, f(n) is O(nlog n) Thus, f(n) is Θ(nlog n)

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