# 1685. Sum of Absolute Differences in a Sorted Array

## Description

You are given an integer array nums sorted in non-decreasing order.

Build and return an integer array result with the same length as nums such that result[i] is equal to the summation of absolute differences between nums[i] and all the other elements in the array.

In other words, result[i] is equal to sum(|nums[i]-nums[j]|) where 0 <= j < nums.length and j != i (0-indexed).

Example 1:

Input: nums = [2,3,5]
Output: [4,3,5]
Explanation: Assuming the arrays are 0-indexed, then
result[0] = |2-2| + |2-3| + |2-5| = 0 + 1 + 3 = 4,
result[1] = |3-2| + |3-3| + |3-5| = 1 + 0 + 2 = 3,
result[2] = |5-2| + |5-3| + |5-5| = 3 + 2 + 0 = 5.


Example 2:

Input: nums = [1,4,6,8,10]
Output: [24,15,13,15,21]


Constraints:

• 2 <= nums.length <= 105
• 1 <= nums[i] <= nums[i + 1] <= 104

## Solutions

Solution 1: Summation + Enumeration

First, we calculate the sum of all elements in the array $nums$, denoted as $s$. We use a variable $t$ to record the sum of the elements that have been enumerated so far.

Next, we enumerate $nums[i]$. Then $ans[i] = nums[i] \times i - t + s - t - nums[i] \times (n - i)$. After that, we update $t$, i.e., $t = t + nums[i]$. We continue to enumerate the next element until all elements are enumerated.

The time complexity is $O(n)$, where $n$ is the length of the array $nums$. The space complexity is $O(1)$.

• class Solution {
public int[] getSumAbsoluteDifferences(int[] nums) {
// int s = Arrays.stream(nums).sum();
int s = 0, t = 0;
for (int x : nums) {
s += x;
}
int n = nums.length;
int[] ans = new int[n];
for (int i = 0; i < n; ++i) {
int v = nums[i] * i - t + s - t - nums[i] * (n - i);
ans[i] = v;
t += nums[i];
}
return ans;
}
}

• class Solution {
public:
vector<int> getSumAbsoluteDifferences(vector<int>& nums) {
int s = accumulate(nums.begin(), nums.end(), 0), t = 0;
int n = nums.size();
vector<int> ans(n);
for (int i = 0; i < n; ++i) {
int v = nums[i] * i - t + s - t - nums[i] * (n - i);
ans[i] = v;
t += nums[i];
}
return ans;
}
};

• class Solution:
def getSumAbsoluteDifferences(self, nums: List[int]) -> List[int]:
ans = []
s, t = sum(nums), 0
for i, x in enumerate(nums):
v = x * i - t + s - t - x * (len(nums) - i)
ans.append(v)
t += x
return ans


• func getSumAbsoluteDifferences(nums []int) (ans []int) {
var s, t int
for _, x := range nums {
s += x
}
for i, x := range nums {
v := x*i - t + s - t - x*(len(nums)-i)
ans = append(ans, v)
t += x
}
return
}

• function getSumAbsoluteDifferences(nums: number[]): number[] {
const s = nums.reduce((a, b) => a + b);
let t = 0;
const n = nums.length;
const ans = new Array(n);
for (let i = 0; i < n; ++i) {
const v = nums[i] * i - t + s - t - nums[i] * (n - i);
ans[i] = v;
t += nums[i];
}
return ans;
}


• /**
* @param {number[]} nums
* @return {number[]}
*/
var getSumAbsoluteDifferences = function (nums) {
const s = nums.reduce((a, b) => a + b);
let t = 0;
const n = nums.length;
const ans = new Array(n);
for (let i = 0; i < n; ++i) {
const v = nums[i] * i - t + s - t - nums[i] * (n - i);
ans[i] = v;
t += nums[i];
}
return ans;
};


• public class Solution {
public int[] GetSumAbsoluteDifferences(int[] nums) {
int s = 0, t = 0;
foreach (int x in nums) {
s += x;
}
int n = nums.Length;
int[] ans = new int[n];
for (int i = 0; i < n; ++i) {
int v = nums[i] * i - t + s - t - nums[i] * (n - i);
ans[i] = v;
t += nums[i];
}
return ans;
}
}