Problem Description

The problem is asking for the minimum possible product sum of two equal-length arrays. The product sum is the sum of a[i] * b[i] for all possible i, given 0-indexed arrays a and b. The twist in the problem is that while one of the arrays (nums2) must remain in its original order, the other array (nums1) can be rearranged in any order to minimize the product sum.

To provide an example, let's assume we have a = [1,2,3,4] and b = [5,2,3,1]. If we were not allowed to rearrange a, then the product sum would be 1*5 + 2*2 + 3*3 + 4*1 = 22. However, if we can rearrange a, we would try to pair the largest numbers in b with the smallest numbers in a to get the smallest possible product for each pair, thereby minimizing the total product sum.

Intuition

To find the minimum product sum, we should try to multiply the smallest numbers in nums1 with the largest numbers in nums2, and vice versa. This is because a large number times a small number adds less to the product sum than a large number times another large number.

To implement this, both arrays nums1 and nums2 are sorted. For nums1, we sort it in non-decreasing order. This places the smallest elements at the start of the array. We do not need to sort nums2 as its order cannot be changed, but for the purpose of the solution, we sort it in non-decreasing order as well.

We then iterate over the arrays, multiplying the i-th element of nums1 with the (n - i - 1)-th element of nums2 which corresponds to the i-th largest element in nums2 because nums2 is sorted in ascending order. The sum of these products gives the minimum product sum possible by this rearrangement.

This method works because sorting nums1 and pairing its elements with the reverse-sorted elements of nums2 ensures that each product added to the sum is as small as it can possibly be considering the constraints of the problem.

Learn more about Greedy and Sorting patterns.

Solution Approach

The solution involves a few steps as described below:

1. Sort the nums1 array in non-decreasing order, which will place the smallest elements at the beginning.
2. Sort the nums2 array in non-decreasing order as well, though its original order does not need to be changed for the problem statement. This step is purely for ease of implementation to allow us to iterate from one end of nums1 to the other while moving in the opposite direction in nums2.

By completing these sorting steps, we can now pair the smallest elements in nums1 with the largest in nums2, which is the key to minimizing the product sum.

3. Initialize a variable res to store the cumulative product sum.
4. Iterate over the length of either nums1 or nums2 (since they are of equal length) using a for loop. Calculate the product by taking the i-th element from the sorted nums1 and the element at index (n - i - 1) from the sorted nums2 array. This effectively reverses the second array during the multiplication process.
5. Add this product to the variable res to keep a running total.
6. After the loop concludes, res contains the minimum product sum, as per the requirements of the problem statement.

The algorithm used is sorting, which under the hood, depending on the language and its implementation, can be a quicksort, mergesort, or similar O(n log n) sorting algorithm. The rest of the function simply iterates through the arrays, which is an O(n) operation.

Here is the code breakdown:

• nums1.sort(): Sorts the first list in non-decreasing order.
• nums2.sort(): Sorts the second list in non-decreasing order.
• n: Captures the length of the arrays.
• res: The variable that will accumulate the total product sum.
• for i in range(n): Iterates through each index of the arrays.
• res += nums1[i] * nums2[n - i - 1]: Calculates the product of the smallest element in nums1 with the largest remaining element in nums2 and adds it to res.

After going through all elements, we simply return res which now holds the minimum product sum.

class Solution:
    def minProductSum(self, nums1: List[int], nums2: List[int]) -> int:
        nums1.sort()
        nums2.sort()
        n, res = len(nums1), 0
        for i in range(n):
            res += nums1[i] * nums2[n - i - 1]
        return res

This is a classic greedy approach that ensures the products of pairs are as small as possible to achieve the global minimum sum.

Example Walkthrough

Let's consider two sample arrays nums1 = [4, 3, 2, 1] and nums2 = [1, 2, 3, 4]. Our goal is to minimize the product sum of these two arrays, with the possibility of rearranging nums1 but keeping the order of nums2 fixed.

Following the solution approach:

1. Sort nums1 in non-decreasing order to get [1, 2, 3, 4].

2. Sort nums2 in non-decreasing order for the purpose of calculation to get [1, 2, 3, 4]. Remember that the order of nums2 isn't actually changed in the final answer—this step is simply to help visualize the process.

3. Initialize res to 0 which will hold the cumulative product sum.

4. Iterate over the arrays, calculating the product sum using the greedy approach:

   For i = 0 (first pairing):

   • The product of nums1[0] and nums2[4 - 0 - 1] (the last element of nums2) is 1 * 4 = 4.

   For i = 1 (second pairing):

   • The product of nums1[1] and nums2[4 - 1 - 1] (the second last element of nums2) is 2 * 3 = 6.

   For i = 2 (third pairing):

   • The product of nums1[2] and nums2[4 - 2 - 1] (the third last element of nums2) is 3 * 2 = 6.

   For i = 3 (fourth pairing):

   • The product of nums1[3] and nums2[4 - 3 - 1] (the first element of nums2) is 4 * 1 = 4.

5. The cumulative sum of these products is 4 + 6 + 6 + 4 = 20, which is stored in res.

6. After completing the iteration, res now contains the minimum product sum which, in this case, is 20.

This demonstrates the solution approach effectively: by sorting nums1 and pairing each element with the 'opposite' element from nums2 (i.e., the element as far from it as possible in the sorted version of nums2), we achieve the minimum product sum. The implemented Python code will return 20 as the result for this example.

Solution Implementation

Time and Space Complexity

Time Complexity

The primary operations in the code are the sorting of nums1 and nums2, followed by a single pass through the arrays to calculate the product sum.

• Sorting: The sort() function in Python typically uses the Timsort algorithm, which has a time complexity of O(n log n). As there are two lists being sorted, this operation is performed twice.

• Single Pass Calculation: After sorting, the code iterates through the lists once, with a single loop performing n iterations, where n is the length of nums1 (and nums2). The operations inside the loop are constant time, so this is O(n).

The overall time complexity is the sum of the two operations, but since O(n log n) dominates O(n), the total time complexity of the algorithm is O(n log n).

Space Complexity

The space complexity of the code is determined by the additional space required apart from the input:

• No extra space is used for storing intermediate results; only a fixed number of single-value variables (n, res) are utilized.
• The sorting algorithms may require O(n) space in the worst case for manipulating the data.

Therefore, the space complexity is O(n).

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