Big-O notation

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This article looks at the definition of Big-O notation, how it works and provides some code examples of different Big-O time complexities.

Introduction

Big-O notation (pronounced ‘Big Oh’) is used in computer science as a means to describe the worst-case performance of an algorithm. It’s one of the things you really should learn if you’re interested at all in designing efficient algorithms.

The formal definition is as follows:

f(n) = O(g(n)) means cg(n) is an upper bound on f(n). Thus there exists some constant c such that f(n) is always \leq cg(n), for large enough n (i.e. n \geq n_0 for some constant n_0).

The Algorithm Design Manual, Steven S. Skiena

This definition is a very formal way of saying that Big-O notation is the upper-bound/worst-case of a function.

Worst-case

Big-O notation has a counterpart Ω-notation (Big Omega notation), which describes the best-case instead of the worst-case. While it may seem unintuitive, this is actually much less useful when analysing performance issues. In general we care much more about the worst-case performance as we want everything to run as smooth as possible, not randomly fast and randomly slow. Consider the following function:

public Boolean doesTextContainChar(String text, char character) {
for (int i = 0; i < text.length(); i++) {
if (text.charAt(i) == character) {
return true;
}
}
return false;
}


If we run this function when text is very small or when character is near the beginning of text then it will run very quickly. But what about when text is extremely long and character doesn’t exist? This is the worst-case of the algorithm.

Simplicity

Big-O notation ignores the details of an algorithm and focuses only on the largest power in the term. A precise analysis of an algorithm could look something like this:

f(n) = 3n^3 + 60n^2 + 8n + 103

The beauty of Big-O notation is that we can ignore most of this and just focus on the largest term. We can even ignore the constant, consider the same algorithm written in a low-level language runs 3 times faster than when written in a high-level language, it’s a detail that isn’t really important when looking at the algorithm itself. So the function above in Big-O notation is

f(n) = O(n^{3})

O(1)

An O(1) function always runs in the same time regardless of the input size.

public Boolean isCharCaps(char character) {
int asciiCode = (int)character;
return asciiCode >= 65 && asciiCode <= 90;
}


O(\log n)

An O(\log n) function grows at a rate of the logarithm (base 2) of the input size. A lot of sorted array search algorithms like the binary search algorithm are O(\log n). This is only really possible when you can ignore input elements entirely.

public Boolean binarySearch(int[] sortedArray, int value, int min, int max) {
if (max < min)
return false;
else {
int mid = (min + max) / 2;

if (sortedArray[mid] > value)
return BinarySearch(sortedArray, value, min, mid-1);
else if (sortedArray[mid] < value)
return BinarySearch(sortedArray, value, mid+1, max);
else
return true;
}
}


O(n)

An O(n) function grows at the same rate as the input size.

public Boolean doesTextContainChar(String text, char character) {
for (int i = 0; i < text.length(); i++) {
if (text.charAt(i) == character) {
return true;
}
}
return false;
}


O(n^2)

An O(n^2) function grows at a rate of input size to the power of 2.

public static int[] insertionSort(int[] array) {
for (int i = 1; i < array.length; i++) {
int item = array[i];
int indexHole = i;
while (indexHole > 0 && array[indexHole - 1] > item) {
array[indexHole] = array[--indexHole];
}
array[indexHole] = item;
}
return array;
}


O(2^n)

An O(2^n) function’s time doubles whenever the input grows. If your function is O(2^n) it’s going to really struggle when the size of your input data increases even a little. Writing one would make me feel a little icky.

Textbooks

Here are two CS textbooks I personally recommend; the Algorithm Design Manual (Steven S. Skiena) is a fantastic introduction to data structures and algorithms without getting to deep into the maths side of things, and Introduction to Algorithms (CLRS) which provides a much deeper, math heavy look.