Big-Omega Notation

Tags: #◇CCDSALG #◇Algorithms #◇Computer Science #◇Computational Complexity #◇Efficiency #◇Algorithm Design Paradigms #◇Data Structures #◇Time Complexity #◇Space Complexity

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◇Overview

f(n) = Ω(g(n)) means f(n) grows at least as fast as g(n).

◇Details / Theory

Asymptotic Lower Bound

f(n) = Ω(g(n)) means f(n) grows at least as fast as g(n). It represents the Best-case running time of the algorithm. Formal Definition: Ω(g(n)) = { f(n) | ∃ c > 0, n₀ > 0 such that 0 ≤ c⋅g(n) ≤ f(n) for all n ≥ n₀ } (Handout 03, Page 4)
Example: 6n³ + 2n² + 3n + 8 is Ω(n³). Also Ω(n²), Ω(n), etc. We usually seek the tightest lower bound. The function $6n^3+2n^2+3n+8$ grows at least as fast as $n^3$, thus the function is $\Omega (n^3)$
Is it also $\Omega(n^4)$? No
Is it $\Omega(n^2)$? $\Omega(n)$? Yes The function is $\Omega(n^c),c\leq n$

◇Complexity

◇Common Pitfalls

Confusing Big-Omega with best-case — Big-Omega is a mathematical bound, not an analysis type. Forgetting the tightest lower bound is preferred: $\Omega(n^2)$ is more informative than $\Omega(1)$.

◇Summary

TL;DR — Big-Omega ($\Omega$) gives an asymptotic lower bound: $f(n) = \Omega(g(n))$ means $f(n)$ grows at least as fast as $g(n)$. Combined with Big-O, it establishes tight bounds.

◇Links

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◇References & Navigation

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• Related Notes: ◇Big-O Notation, ◇Big-Theta Notation