How do you compare algorithm growth rates?

1 Why growth rates matter

The growth rate of an algorithm is a function that describes how the running time or space usage of the algorithm changes as the input size increases. For example, if an algorithm takes n steps to sort n elements, its growth rate is O(n). If it takes n^2 steps, its growth rate is O(n^2). The growth rate gives you an idea of how scalable and reliable an algorithm is, and how it behaves under different scenarios. It also helps you to compare algorithms based on their worst-case, average-case, or best-case performance.

2 Big O notation

The most common way to express the growth rate of an algorithm is using the big O notation. The big O notation captures the upper bound of the growth rate, meaning that the algorithm will not take more time or space than the function inside the big O. For example, if an algorithm has a growth rate of O(n), it means that it will not take more than n steps for any input of size n. The big O notation ignores the constant factors and lower-order terms that do not affect the asymptotic behavior of the algorithm. For example, O(2n+5) is equivalent to O(n), and O(n^2+3n+1) is equivalent to O(n^2).

3 Big Omega notation

The big Omega notation is the counterpart of the big O notation. It captures the lower bound of the growth rate, meaning that the algorithm will not take less time or space than the function inside the big Omega. For example, if an algorithm has a growth rate of Omega(n), it means that it will take at least n steps for any input of size n. The big Omega notation also ignores the constant factors and lower-order terms. For example, Omega(0.5n-2) is equivalent to Omega(n), and Omega(n^2-4n+2) is equivalent to Omega(n^2).

4 Big Theta notation

The big Theta notation is a combination of the big O and big Omega notations. It captures the tight bound of the growth rate, meaning that the algorithm will take exactly the same time or space as the function inside the big Theta, up to a constant factor. For example, if an algorithm has a growth rate of Theta(n), it means that it will take c*n steps for some constant c and any input of size n. The big Theta notation also ignores the constant factors and lower-order terms. For example, Theta(3n+7) is equivalent to Theta(n), and Theta(n^2+2n-5) is equivalent to Theta(n^2).

5 How to compare growth rates

To compare the growth rates of different algorithms, you can use the following rules. If two algorithms have the same growth rate, they are equally efficient asymptotically, such as O(n) and O(2n). If one algorithm has a lower growth rate than another, it is more efficient asymptotically; for example, O(n) is better than O(n^2). Additionally, if one algorithm has a growth rate that is a subset of another, it is more efficient asymptotically; for example, O(n) is better than O(n log n), because it is a subset of the latter. Finally, if two algorithms have different growth rates that are not subsets of each other, you can compare them by taking the limit of their ratio as n approaches infinity. For instance, you can compare O(n^2) and O(2^n) by taking the limit of n^2/2^n as n approaches infinity, which is zero. This means that O(n^2) is better than O(2^n).

6 Examples of growth rates

Growth rates and their corresponding algorithms can vary greatly. O(1) is a constant time algorithm, meaning it takes the same time or space regardless of the input size. An example of this would be accessing an element in an array by its index. O(log n) is a logarithmic time algorithm, which reduces the input size by a constant factor in each step. An example of this is binary search in a sorted array. O(n) is linear time, meaning it processes each element in the input once; finding the maximum element in an array being an example. O(n log n) is linearithmic time, combining a linear and a logarithmic operation in each step, such as merge sort or quick sort. O(n^2) is quadratic time, processing each pair of elements in the input like bubble sort or insertion sort. O(n^3) is cubic time and processes each triple of elements in the input like matrix multiplication. O(2^n) is exponential time, doubling the input size in each step; finding all the subsets of a set being an example. Finally, O(n!) is factorial time, increasing the input size by a factorial in each step; for instance finding all the permutations of a set.

7 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?