Simplify function to sum the absolute value of the difference between all pairs of elements of a sorted array

Question

Given a sorted array of N elements, I need to find the absolute sum of the differences of all elements.

Given 4 elements (1, 2, 3 and 4):

|1-2|+|1-3|+|1-4|+|2-3|+|2-4|+|3-4| = 10

Here is my code in Java:

List<Integer> a = new ArrayList<Integer>(); //just for understanding , the Array List is already filled with numbers 
public static int lsum(int N)
{
    int sum =0;

    for( int i=0;i<N;i++)
    {
        int w =a.get(i);
        for(int j =i;j<N;j++)
        {
            int z = a.get(j);
            sum =sum +(z-w);
        }

        }
    return(sum);

}

I'm looking for an efficient algorithm rather than the trivial one I am using (O(n²) complexity). This is a requirement for a bigger program which requires this function. The input (number of elements) can be as big as 10⁵.

Answer 1

Let's see this mathematically. I am assuming that a is an ascendingly sorted array. I start with indexes from 1, to get better readability (by avoiding n-1 as often as possible).

We want the following sum:

$$\begin{array}{l@{}l@{}l@{}l} \sum_{i=1}^{n-1} \sum_{j=i+1}^n (a_j - a_i) = (a_2 - a_1) &+ (a_3 - a_1) &+ (a_4 - a_1) + & \dots + (a_n - a_1) +\\ &+ (a_3 - a_2) &+ (a_4 - a_2) + & \dots + (a_n - a_2) +\\ &&+ (a_4 - a_3) + & \dots + (a_n - a_3) +\\ &&& \dots \end{array}$$

So, we:

1. add a1 zero times, a2 once,... ak k-1 times, and
2. subtract a1 n-1 times, a2 n-2 times,... ak n-k times.

I'd say that your sum is

$$ \sum_{k=1}^n (k-1)a_k - \sum_{k=1}^n (n-k)a_k = \sum_{k=1}^n (2k-n-1)a_k $$

Given your example (1,2,3,4), we get:

$$ \sum_{k=1}^4 (2k-4-1)a_k = -3 \cdot 1 + (-1 \cdot 2) + 1 \cdot 3 + 3 \cdot 4 = -3 - 2 + 3 + 12 = 10. $$

This has a linear complexity, which is optimal, because you need to "visit" each member of the array at least once (which is linear).

I believe you can write your code by yourself. Just set indexes to go from zero, but be careful about the formula: 2k - n - 1 becomes 2k - n + 1 when k starts from zero.

Answer 2

I would not give you the exact solution to this problem rather I would help you to get an intuition on how to solve this question.

If we try to generalize count of the number of times a particular number at index i is getting added and the number of times it is being subtracted then for every index i we can use that mathematically derived formula to compute the sum of contributions of every number in the absolute difference in O(N) time and O(1) extra space.

You can refer to this video if you still find difficulties in understanding this concept.

Video Link:-

https://youtu.be/A4sz4kNDa-8