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Algorithm Analysis: Big-Oh Notation

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Questions and Answers

What is the function of the Big-Oh notation in algorithm analysis?

It provides an exact count of primitive operations.

It measures the memory usage of algorithms.

It categorizes algorithms by their input data type.

It characterizes the main factors affecting an algorithm’s running time. (correct)

Under what conditions can we say that f(n) is O(g(n))?

If f(n) is strictly less than g(n) for all n.

If f(n) is always greater than g(n) for large values of n.

If f(n) grows at a constant rate independent of g(n).

If there exist constants c > 0 and n0 such that f(n) ≤ c g(n) for all n ≥ n0. (correct)

In the proof that 7n - 2 is O(n), what constants can be chosen to satisfy the definition of Big-Oh?

c = 8 and n0 = 3

c = 6 and n0 = 1

c = 5 and n0 = 2

c = 7 and n0 = 1 (correct)

Which of the following statements correctly describes a characteristic of pseudocode in algorithm analysis?

<p>Each step tends to correspond to a small number of primitive operations that do not depend on input size.</p> Signup and view all the answers

What is the meaning of 'c > 0' in the definition of Big-Oh notation?

<p>It is a constant that scales g(n) to assess the upper bound of f(n).</p> Signup and view all the answers

Study Notes

Asymptotic Notation

A method for evaluating the running time of algorithms that simplifies analysis by focusing on the main factors affecting the running time.
Aims to estimate the number of primitive operations executed up to a constant factor, regardless of the specific number of operations per statement.

Big-Oh Notation

Used to characterize the running time of algorithms.
Defined as follows:
- Let f ( n ) and g ( n ) be functions mapping nonnegative integers to real numbers.
- We say that f ( n ) is O ( g ( n ) ) if there is a real constant c > 0 and an integer constant n 0 ≥ 1 such that f ( n ) ≤ cg ( n ) for every integer n ≥ n 0.
Often pronounced as “ f ( n ) is big-Oh of g ( n ) ” or “ f ( n ) is order g ( n ).”

Example

7 n − 2 is O ( n ) because we can find a real constant c > 0 and an integer constant n 0 ≥ 1 such that 7 n − 2 ≤ cn for every integer n ≥ n 0.
- For example, c = 7 and n 0 = 1 satisfies the condition.

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Description

This quiz covers Big-Oh notation, a critical method for analyzing the running time of algorithms. It simplifies the evaluation of algorithm efficiency by focusing on the dominant factors affecting performance. Test your understanding of its definitions and applications with this quiz.