CS 411 Fall 2025  >  Outline for August 29, 2025

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CS 411 Fall 2025
Outline
for August 29, 2025

Outline

Asymptotic Notation [L 2.2]

• Big-\(O\) notation
  • Used to express order of growth
  • Is about a function. Most common function is one that gives maximum number of steps required for a given input size.
• \(\Omega\) and \(\Theta\) notations
• Property: this-then-that
  • Order of sum is max of orders.
    • Example: \(n+n^2\) is \(\Theta(n^2)\).
  • So if an algorithm does this and then that, the order of growth of the number of steps required is the max of the order of this & the order of that.
• Using limits
  • To determine whether \(f(n)\) or \(g(n)\) is of higher order, we can compute

    \[ \lim_{n\to+\infty} \frac{f(n)}{g(n)}. \]

    • If the result is \(0\), then \(g(n)\) is of higher order.
    • If the result is \(+\infty\), then \(f(n)\) is of higher order.
    • If the result is a constant \(c > 0\), then \(f(n)\) and \(g(n)\) are of the same order.
    L’Hôpital’s Rule often helps.
  • When dealing with factorials, the computations are often simpler if we use Stirling’s formula:

    \[ n! \approx \frac{n^n}{e^n}\sqrt{2\pi n}. \]

    This is a good approximation for the factorial in the following sense:

    \[ \lim_{n\to+\infty} \frac{\frac{n^n}{e^n}\sqrt{2\pi n}}{n!} = 1. \]

    Because of this, the approximation to the factorial given by Stirling’s formula is more than good enough for order-of-growth computations.
• Classes (small to large; the base of logarithms does not matter)
  • \(1\): constant
  • \(\log n\): logarithmic
  • \(n\): linear
  • \(n\log n\): linearithmic, a.k.a. log-linear
  • \(n^2\): quadratic
  • \(n^3\): cubic
  • \(c^n\) for a constant \(c > 0\): exponential
  • \(n!\): factorial