Let's go through a few applications.

h2. 4sum (popular interview question)

h2. 4sum

bq. Given A, an array of integers, find out if there are any four numbers in the array that sum up to zero (the same element can be used multiple times). For example given A = [2, 3, 1, 0, -4, -1] a solution is 3 + 1 + 0 - 4 = 0 or 0 + 0 + 0 + 0 = 0.

bq. Given A, an array of integers, find out if there are any four numbers in the array that sum up to zero (the same element can be used multiple times). For example given A = [2, 3, 1, 0, -4, -1] a solution is 3 + 1 + 0 - 4 = 0 or 0 + 0 + 0 + 0 = 0.
 
4sum is one of the most popular programming interview questions.

The naive algorithm checks all four number combinations. This solution takes $O(N^4^)$ time.