Question

1685. Sum of Absolute Differences in a Sorted Array

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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:
  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

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;
  }
};

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;
  }
}