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Discrete Structures – CNS2300PowerPoint Presentation

Discrete Structures – CNS2300

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Discrete Structures – CNS2300

Text

Discrete Mathematics and Its Applications

Kenneth H. Rosen

Chapter 2

The Fundamentals: Algorithms, the Integers, and Matrices

Section 2.2

The Growth of Functions

Big-O notation

Let f and g be functions from the set of integers or the set of real numbers to the set of real numbers. We say that f(x) is O(g(x)) if there are constants C and k such that

|f(x)| <= C|g(x)|

whenever x > k.

Read f(x) is “big-oh” of g(x).

x f(x) g(x)1/4 5/16 3/16

2/4 25/16 12/16

3/4 33/16 27/16

4/4 48/16 48/16

5/4 65/16 75/16

Consider f(x) = x2 + 2x

g(x) = 3x2

Let C=3

Let k=1

Suppose that f1(x) is O(g1(x)) and f2(x) are both O(g(x)) . Then (f1+f2)(x) isO(g(x)).

The Growth of Combinations of FunctionsSuppose that f1(x) is O(g1(x)) and f2(x) is O(g2(x)) . Then (f1+f2)(x) isO(max(g1(x), g2(x))).

The Growth of Combinations of Functions

Suppose that f1(x) is O(g1(x)) and f2(x) is O(g2(x)) . Then (f1f2)(x) isO( g1(x)g2(x)).

Computer Times Used by Algorithms

n log nnnlog nn2 2nn!

10 3E-9 E-8 3E-8 E-7 E-6 3E-3

102 7E-9 E-7 7E-7 E-5 4E13yr *

103 1E-8 E-6 E-5 E-3 * *

104 1.3E-8 E-5 E-4 E-1 * *

105 1.7E-8 E-4 2E-3 10 * *

106 2E-8 E-3 2E-2 17min * *

All times are in seconds unless otherwise stated. An * indicates a time of more than 10100 years.

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