Big-O notation. Suppose n is the input size, we have the following commonly seen functions in complexity analysis: f1(n) = 1, f2(n) = log n, f3(n) = n, f4(n) = n log n, f5(n) = n2, f6(n) = 2n, f7(n) = n!, f8(n) = nn. Intuitively, the growth rate of the functions satisfy 1 < log n < n < n log n < n2 < 2n < n! < nn. Prove this is true. Let f, g : N → R+, prove that Ω(f(n) + g(n)) = Ω(max{f(n), g(n)}).[Note: Proving this will help you understand that we can also leave out the insignificant parts in big-Ω notation and the result is still a lower bound, e.g., Ω(n2 + n log n + n) = Ω(n2).]

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Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

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Big-O notation.

Suppose n is the input size, we have the following commonly seen functions in complexity analysis: f₁(n) = 1, f₂(n) = log n, f₃(n) = n, f₄(n) = n log n, f₅(n) = n², f₆(n) = 2ⁿ, f₇(n) = n!, f₈(n) = nⁿ. Intuitively, the growthrate of the functions satisfy 1 < log n < n < n log n < n² < 2ⁿ < n! < nⁿ. Prove this is true.

Let f, g : N → R+, prove that Ω(f(n) + g(n)) = Ω(max{f(n), g(n)}).

[Note: Proving this will help you understand that we can also leave out the insignificant parts in big-Ω notation and the result is still a lower bound, e.g., Ω(n² + n log n + n) = Ω(n²).]

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