You are given two integer arrays nums1 and nums2 of equal length n and an integer k. You can perform the following operation on nums1:
Choose two indexes i and j and increment nums1[i] by k and decrement nums1[j] by k. In other words, nums1[i] = nums1[i] + k and nums1[j] = nums1[j] - k.
nums1 is said to be equal to nums2 if for all indices i such that 0 <= i < n, nums1[i] == nums2[i].
Return the minimum number of operations required to make nums1 equal to nums2. If it is impossible to make them equal, return -1.
Example 1:
Input: nums1 = [4,3,1,4], nums2 = [1,3,7,1], k = 3
Output: 2
Explanation: In 2 operations, we can transform nums1 to nums2.
1st operation: i = 2, j = 0. After applying the operation, nums1 = [1,3,4,4].
2nd operation: i = 2, j = 3. After applying the operation, nums1 = [1,3,7,1].
One can prove that it is impossible to make arrays equal in fewer operations.
Example 2:
Input: nums1 = [3,8,5,2], nums2 = [2,4,1,6], k = 1
Output: -1
Explanation: It can be proved that it is impossible to make the two arrays equal.
Constraints:
n == nums1.length == nums2.length
2 <= n <= 105
0 <= nums1[i], nums2[j] <= 109
0 <= k <= 105
Solutions
Solution 1: Single Pass
We use a variable $x$ to record the difference in the number of additions and subtractions, and a variable $ans$ to record the number of operations.
We traverse the array, and for each position $i$, if $k=0$ and $a_i \neq b_i$, then it is impossible to make the two arrays equal, so we return $-1$. Otherwise, if $k \neq 0$, then $a_i - b_i$ must be a multiple of $k$, otherwise it is impossible to make the two arrays equal, so we return $-1$. Next, we update $x$ and $ans$.
Finally, if $x \neq 0$, then it is impossible to make the two arrays equal, so we return $-1$. Otherwise, we return $\frac{ans}{2}$.
The time complexity is $O(n)$, and the space complexity is $O(1)$, where $n$ is the length of the array.