Characterizing Running Times

This post extracts some knowledge from Introduction to Algorithms Chapter 3 -- Characterizing Running Times.

Asymptotic notation

Asymptotic notation describes how a running time grows as the input size $n$ becomes large. It ignores constant factors and lower-order terms, so we can focus on the main growth trend.

Big-O notation

$O(g(n))$ gives an asymptotic upper bound.

If $f(n) = O(g(n))$, then $f(n)$ grows no faster than $g(n)$, up to a constant factor.

Example: $3n^2 + 10n + 5 = O(n^2)$.

Omega notation

$\Omega(g(n))$ gives an asymptotic lower bound.

If $f(n) = \Omega(g(n))$, then $f(n)$ grows at least as fast as $g(n)$, up to a constant factor.

Example: $3n^2 + 10n + 5 = \Omega(n^2)$.

Theta notation

$\Theta(g(n))$ gives an asymptotic tight bound.

If $f(n) = \Theta(g(n))$, then $f(n)$ grows at the same rate as $g(n)$, up to constant factors.

Example: $3n^2 + 10n + 5 = \Theta(n^2)$.

Little-o notation

$o(g(n))$ gives an upper bound that is not tight.

If $f(n) = o(g(n))$, then $f(n)$ grows strictly slower than $g(n)$.

Example: $n = o(n^2)$.

Little-omega notation

$\omega(g(n))$ gives a lower bound that is not tight.

If $f(n) = \omega(g(n))$, then $f(n)$ grows strictly faster than $g(n)$.

Example: $n^2 = \omega(n)$.

Quick comparison

Notation	Meaning	Intuition
$O(g(n))$	upper bound	at most this fast
$\Omega(g(n))$	lower bound	at least this fast
$\Theta(g(n))$	tight bound	exactly this growth rate
$o(g(n))$	strict upper bound	slower than this
$\omega(g(n))$	strict lower bound	faster than this

Common growth order

From slower to faster growth:

$$ 1 < \log n < n < n \log n < n^2 < n^3 < 2^n < n! $$

The running time with the slower growth rate is usually better for large inputs.

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原文链接：https://zhuang.dev/reading/introduction-to-algorithms/03-characterizing-running-times/