zeinab eidalgorithm analysis1 chapter 4 analysis tools

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Chapter 4

Analysis Tools

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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.

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4.1 Seven Functions

1. The Constant Function: f(n)= c,

where c is some fixed constant, c=5, c=100, or c=210, so we let g(n)=1, so f(n)=cg(n)

2. The Logarith Function: f(n)= logbn

x = logbn iff bx=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).

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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)= n2

It’s important for algorithms of nested loops.

6. The Cubic Function (Polynomial): f(n)= n3

f(n)=a0+a1n+ a2n2+…+adnd, is a polynomial of degree d,

where a0 , a1 ,…., ad are constants. 7. The Exponential Function: f(n)= bn

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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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Primitive Operations

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Counting Primitive Operations

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4.2.3 Asymptotic Notation

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Example

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Properties of Big-Oh Notation

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Big-Omega and Big-Theta

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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 n0 ≥ 1 such that:

f(n) ≥ cg(n), for n ≥ n0

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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 n0 ≥ 1 such that:

c’g(n) ≤ f(n) ≥ c”g(n), for n ≥ n0 .

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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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