Sum of Absolute Differences in a Sorted Array

class Solution {
public:
    vector<int> calculateDifferenceSums(vector<int>& elements) {
        int count = elements.size();
        int sumAll = accumulate(elements.begin(), elements.end(), 0);
            
        int sumLeft = 0;
        vector<int> result;
        for (int i = 0; i < count; i++) {
            int sumRight = sumAll - sumLeft - elements[i];
                
            int countLeft = i;
            int countRight = count - 1 - i;
                
            int totalLeft = countLeft * elements[i] - sumLeft;
            int totalRight = sumRight - countRight * elements[i];
                
            result.push_back(totalLeft + totalRight);
            sumLeft += elements[i];
        }
            
        return result;
    }
};

This solution involves computing the sum of absolute differences in a sorted array. Here's a breakdown of how the provided C++ code achieves this:

• Initialize necessary variables:

  • count to store the number of elements in the input array.
  • sumAll to calculate the sum of all elements in the array using std::accumulate.
  • sumLeft to keep a running sum of the elements processed so far from the left.
  • An empty result vector to store the results.

• Iterate through each element of the array:

  • Calculate sumRight, which is the sum of elements to the right of the current element.
  • Determine countLeft and countRight, representing the number of elements on the left and right of the current element, respectively.
  • Compute totalLeft as the difference between countLeft times the current element and sumLeft.
  • Calculate totalRight as the difference between sumRight and countRight times the current element.
  • Add the sum of totalLeft and totalRight to the result vector.
  • Update sumLeft by adding the current element to it.

• Return the result vector containing the calculated values.

The core logic revolves around leveraging the sorted nature of the array to efficiently compute the sums of absolute differences for each element relative to all other elements. By keeping track of the sums and counts of elements to the left and right of the current element, the solution ensures optimal performance.

class Solution {
    public int[] calculateAbsoluteDifferences(int[] elements) {
        int elementCount = elements.length;
        int sumAllElements = 0;
        for (int element : elements) {
            sumAllElements += element;
        }
            
        int sumPreviousElements = 0;
        int[] result = new int[elementCount];
        for (int index = 0; index < elementCount; index++) {
            int sumFollowingElements = sumAllElements - sumPreviousElements - elements[index];
                
            int previousCount = index;
            int followingCount = elementCount - index - 1;
                
            int sumLeftDifference = previousCount * elements[index] - sumPreviousElements;
            int sumRightDifference = sumFollowingElements - followingCount * elements[index];
                
            result[index] = sumLeftDifference + sumRightDifference;
            sumPreviousElements += elements[index];
        }
            
        return result;
    }
}

The provided solution details a method for computing the sum of absolute differences for each element in a sorted array. Here's a breakdown of how the method functions:

1. Initialize elementCount to ascertain the number of elements in the array.

2. Calculate sumAllElements which is the total sum of all the elements in the array.

3. Initialize sumPreviousElements to keep track of the sum of elements before the current index as the iterations proceed.

4. Create an array result to store the results for each element.

5. Use a loop to iterate through the array. For each element at the specified index:

   • Calculate sumFollowingElements which is the sum of the elements after the current index.
   • previousCount and followingCount respectively count the number of elements before and after the current index.
   • Compute sumLeftDifference, which is the difference multiplied by the count of previous elements, minus sumPreviousElements.
   • Calculate sumRightDifference, which is sumFollowingElements less the product of followingCount times the current element.
   • Sum the left and right differences and assign this value to the respective index in the result array.
   • Update sumPreviousElements by adding the current element.

6. Return the result array containing the desired sums of absolute differences for each element.

This algorithm efficiently leverages cumulative sums to calculate the required differences, ensuring each element’s result depends on previously computed totals, thus optimizing the performance by avoiding redundant recalculations.

class Solution:
    def calcSumOfAbsDifferences(self, elements: List[int]) -> List[int]:
        size = len(elements)
        total_sum = sum(elements)
        sum_to_left = 0
        result = []
    
        for index in range(len(elements)):
            sum_to_right = total_sum - sum_to_left - elements[index]
                
            count_to_left = index
            count_to_right = size - 1 - index
                
            sum_left = count_to_left * elements[index] - sum_to_left
            sum_right = sum_to_right - count_to_right * elements[index]
    
            result.append(sum_left + sum_right)
            sum_to_left += elements[index]
            
        return result

The given Python solution calculates the sum of absolute differences for each element in a sorted array. The function calcSumOfAbsDifferences takes a list of integers (elements) and returns a new list with the computed result. Below, understand each step that this function executes:

1. Determine the length of elements and calculate the total sum of all elements.
2. Initialize sum_to_left to zero and prepare an empty list result for storing the final output.
3. Iterate over all elements in the input list to compute the sum of absolute differences for each element:
   • Calculate the sum to the right of the current element by subtracting sum_to_left and the current element from the total_sum.
   • Define count_to_left as the current index and count_to_right as the number of elements to the right of the current element.
   • Compute sum_left, the difference between the product of count_to_left and the current element and the sum_to_left.
   • Compute sum_right, the difference between sum_to_right and the product of count_to_right and the current element.
   • Append the sum of sum_left and sum_right to the result list.
   • Update sum_to_left by adding the current element to it.
4. Return the result list containing the sum of absolute differences for each element.

This implementation efficiently uses a single loop to manage the calculations, leveraging the properties of sorted arrays to optimize the computation of differences.