Lecture 24: Asymptotic analysis & big O notation

lecture video

Textbook chapters

4.3

Moving beyond experimental analysis: theoretical analysis of algorithms

  • Counting primitive operations
  • Measuring operations as a function of input size
  • Focusing on worst-case input

Big-O notation

Given functions $f(n)$ and $g(n)$, we say that $f(n)$ is $O(g(n))$ if there are positive constants $c$ and $n_0$ such that

\[f(n) \leq cg(n)\]

for $n \geq n_0$.

Some Big-O rules and intuitions

  • all primitive operations run in $O(1)$ time
  • lower-order terms and constants don’t matter
  • we don’t have to give tight bounds ($n$ is $O(n^3)$)
  • big-O does not refer only to worst-case runtimes

Files used in class

Additional problems

  • Order the following functions by their asymptotic growth rate: $4n \log n + 2n$, $2^{10}$, $2^{\log n}$, $3n + 100 \log n$, $4n$, $2^n$, $n^2 + 10n$, $n^3$, $n \log n$.
  • Give the big-O characterization, in terms of $n$, of the runtime of the following algorithm.
    public static int example(int[] arr) {
  int n = arr.length; total = 0;
  for (int j = 0; j < n; j+= 2) { // note the increment of 2
      total += arr[j];
  }
}