Problem Description

In this problem, we are working with two integer arrays, nums1 and nums2, both of the same length n. We're given an integer k which we can use in a specific operation that we can perform on nums1 as many times as needed. An operation consists of picking two indices, i and j, in nums1 and adding k to the value at index i while subtracting k from the value at index j. Our goal is to make the elements of nums1 match the corresponding elements in nums2 using the least number of these operations. If it is impossible to make the arrays equal, we must return -1.

To understand whether it's possible to make nums1 equal to nums2, we should consider that each operation effectively transfers k units of value from one element of nums1 to another. Therefore, if the difference between any pair of corresponding elements in nums1 and nums2 is not a multiple of k, we cannot make the elements equal through the allowed operation, and we should return -1. If all differences are multiples of k, we can then calculate the minimal number of operations required by summing the absolute values of the quotients of differences by k.

Intuition

The intuition behind the solution is to first iterate over each corresponding pair of elements between nums1 and nums2. We calculate the difference between the elements of nums1 and nums2. If k is zero, we immediately know that unless all elements are already equal, it's impossible to make them equal as no operation can modify nums1. If k is nonzero, any difference that isn't a multiple of k would make it impossible for the two corresponding elements to ever match, so we return -1.

If a difference is a multiple of k, this multiple represents how many operations of magnitude k are needed to equalize this pair of elements. We accumulate this count in an ans variable.

Additionally, we keep track of the net effect of all operations in a variable x. This is because each operation has two parts: incrementing one element and decrementing another. So, if x is nonzero after considering all pairs, it means we've either incremented or decremented too much and can't balance this out with further operations. If x equals zero, it means that for every increment there is a corresponding decrement, and nums1 can be made equal to nums2.

However, since every operation affects two elements (increments one and decrements another), the total number of operations required is half the sum of absolute values of all quoted differences, as each operation is counted twice in ans. Hence, we return ans // 2, but only if x equals zero, which ensures that all increments and decrements are paired.

Learn more about Greedy and Math patterns.

Solution Approach

The solution follows a straightforward approach where it iterates through the two lists simultaneously using the built-in zip function. For each pair of elements (a, b) from nums1 and nums2 respectively, it performs several checks and calculations.

Here's a step-by-step breakdown of the implementation:

• Initialize ans and x to 0. ans will hold the total number of operations required, while x will track the net effect of all those operations.
• Iterate over nums1 and nums2 in tandem using zip(nums1, nums2).
• For each pair (a, b):
  • Check if k is 0. If it is and a != b, return -1 immediately because no operations can be performed to change the values.
  • Calculate the difference a - b and check if it is a multiple of k. This is done by checking if (a - b) % k equals 0. If it does not, return -1 because it's impossible to make a equal to b through the allowed operation.
  • Otherwise, compute the quotient y = (a - b) // k, which represents how many operations of magnitude k are needed to equalize a and b. We need to consider the absolute value of y to add to ans because operations can be incrementing or decrementing, and we're counting total operations needed regardless of direction.
  • Add y to the net effect tracker x. An increment in one element will be a positive y, and a decrement will be a negative y, so the net effect after all pairs are considered should balance to zero.
• After the loop, check if x is nonzero. If it is, return -1 because the net effect hasn't balanced out, implying that not all increments have a corresponding decrement.
• If x is zero, return ans // 2. Each operation affects two elements of nums1, so every actual operation is counted twice in ans.

This solution approach uses simple mathematical operations and control structures to determine the minimum number of operations. It leverages the fact that each allowed operation can be represented mathematically as an equation and that the collective effect of these operations must balance for equality to be possible.

Example Walkthrough

Let's go through a small example to illustrate the solution approach.

Suppose we have the following input:

• nums1 = [1, 3, 5]
• nums2 = [5, 1, 3]
• k = 2

We want to make each element in nums1 match the corresponding element in nums2 using the smallest number of operations.

Following the solution approach:

1. Initialize ans and x to 0. These will track the number of operations and the net effect, respectively.

2. Iterate over the pairs (1, 5), (3, 1), and (5, 3) from nums1 and nums2.

3. Starting with the first pair (1, 5):

   • The difference 1 - 5 is -4, which is a multiple of k (2).
   • The quotient y is -4 // 2 = -2. We take the absolute value |y| = 2 and add it to ans, so now ans = 2.
   • We add y (which is -2) to x to track the net effect, now x = -2.

4. Next, the pair (3, 1):

   • The difference 3 - 1 is 2, which is a multiple of k.
   • The quotient y is 2 // 2 = 1. We add the absolute value |y| = 1 to ans, so now ans = 3.
   • We add y (which is 1) to x, making the net effect now x = -1.

5. Finally, the pair (5, 3):

   • The difference 5 - 3 is 2, which is also a multiple of k.
   • The quotient y is 2 // 2 = 1. We add the absolute value |y| = 1 to ans, now ans = 4.
   • We add y to x, and as x was previously -1 and y is 1, the net effect balances out, x = 0.

6. Since x is 0, all increments have matching decrements. We can therefore equalize nums1 to nums2.

7. Our answer is ans // 2 = 4 // 2 = 2. Hence, the minimum number of operations required is 2.

This example clearly demonstrated that we need to perform two operations of magnitude k (2 in this case) to make nums1 equal to nums2. The steps as outlined in the solution approach guide us through a process of calculating the necessary number of operations or determining the impossibility of equalizing the two arrays.

Solution Implementation

Time and Space Complexity

The time complexity of the given code is O(N) where N is the length of the shorter of the two input lists, nums1 and nums2. This is because the code iterates over each pair of elements from nums1 and nums2 once by using zip, and the operations within the loop (such as modulo, division, comparison, and addition) are all O(1) operations. No nested loops or recursive calls that would increase the complexity are present.

The space complexity of the code is O(1) since the space used does not increase with the size of the input; it only uses a fixed number of variables (ans, x, a, b, y) to process the input and produce the output.

Learn more about how to find time and space complexity quickly using problem constraints.