Problem Description

You are given two integer arrays nums1 and nums2, both of the same length n and indexed starting from zero. You have the ability to perform operations on these arrays, where a single operation consists of swapping the values at the same index i in both arrays. The goal is to make the last element of each array the maximum value within that respective array. The problem asks to determine the minimum number of such operations required to achieve this goal for both arrays—or to establish that it is impossible to do so, in which case the result should be -1.

To clarify:

• nums1[n - 1] should be equal to the maximum value in nums1.
• nums2[n - 1] should be equal to the maximum value in nums2.

You have to return the smallest number of operations needed to satisfy both conditions or -1 if it's not possible to satisfy them.

Intuition

First, the problem can be approached by considering two cases: swapping or not swapping the values of nums1[n - 1] and nums2[n - 1]. For each case, two scenarios can occur for any index i: either no swap is needed because both elements are already in their correct place according to the maximum value constraint, or a swap is needed to move a larger value to the last index of its respective array.

We can create a helper function f(x, y) that will count the number of operations (swaps) needed when x and y represent the last values of nums1 and nums2 respectively. As we loop through the arrays, if we encounter a pair where neither array can have their last value as the maximum without swapping with the other, we immediately know it is impossible to satisfy the condition, and we can return -1.

The solution employs a greedy strategy:

• If both values at the current index i are less than or equal to x and y respectively, no action is needed.
• If one value is greater than x and can be swapped to be the new last value of nums1, and the other is greater than y and can be swapped to be the new last value of nums2, we perform a swap.
• If neither of these conditions is satisfied, fulfilling the goal is not possible.

After considering both cases of whether to swap the last elements or not, we'll have two values representing the minimum number of swaps needed for each scenario. If both scenarios return -1, this means it's impossible to meet the goal for both arrays. Otherwise, we return the smaller number of operations required, bearing in mind that if we did swap the last elements initially, an extra swap (totaling b + 1) is included in the final count.

Solution Approach

The implementation consists of a helper function f(x, y) that is responsible for determining the number of swaps required to make x the last value of nums1 and y the last value of nums2. This function iterates through the arrays, excluding their last elements, to check if the current condition meeting criteria is satisfied without swapping or if a swap is needed.

Here are the steps in the algorithm:

1. Check each element pair from nums1 and nums2 (up to n-1) to see if they already satisfy the condition where nums1[i] <= x and nums2[i] <= y.
2. If the condition is not satisfied, check if making a swap would satisfy it, which means nums1[i] <= y and nums2[i] <= x after the swap.
3. If neither the current state nor a swap can satisfy the condition, it means it's impossible to achieve the goal, and f(x, y) will return -1.
4. If a swap is needed, the count cnt is incremented.

Following this, the minOperations() method calls f(x, y) twice with different parameters: once where x and y are the initial last values of nums1 and nums2 respectively (case 1: no swap for last elements), and then with x and y swapped (case 2: with swap for last elements). It stores the results in variables a and b.

The final step in the algorithm is to evaluate these two outcomes:

• If both a and b are -1, it means it's impossible to satisfy the conditions using either case, and the method returns -1.
• If at least one of the cases has a non--1 result, the method returns the minimum number of operations needed to satisfy the conditions. Since swapping the last elements counts as an operation, in case 2 (if b is not -1), 1 is added to b before comparing it to a.

The implementation makes use of:

• A helper function to encapsulate the logic for checking and counting swaps needed.
• A greedy approach that incrementally solves the problem by deciding the best action (swap or no swap) for each element pair.
• Iterative traversal of both arrays which efficiently enables us to check conditions and count swaps at the same time.

Example Walkthrough

Let's walk through a small example to illustrate the solution approach. Consider the following arrays:

nums1 = [1,3,5,4]
nums2 = [1,2,3,7]

Both arrays have a length of n = 4. According to the problem, we want nums1[3] to be the maximum of nums1 and nums2[3] to be the maximum of nums2.

Initially, nums1[3] = 4, and nums2[3] = 7. The maximum values of nums1 and nums2 are 5 and 7, respectively. We need to make 5 the last element of nums1 and keep 7 as the last element of nums2.

1. We start with the helper function f(x, y), with x = 4 and y = 7, the last elements of nums1 and nums2 respectively.
2. Loop through the arrays up to n-1 (i.e., the third element), checking if a swap is necessary:
   • Index 0: Both 1 and 1 are less than 4 and 7. No swap needed.
   • Index 1: 3 is less than 4, and 2 is less than 7. No swap needed.
   • Index 2: 5 is greater than 4 and needs to be the last element of nums1, and 3 is less than 7, so we can swap 5 and 4 to satisfy the condition for nums1. After this swap, nums1 becomes [1,3,4,5].
3. Since we were able to make a swap that progresses towards our goal, we increment our count cnt by 1.
4. Now, we call the helper function f(x, y) again with x = 7 and y = 4 to see if we should have swapped the last elements initially.
   • This time, the elements of nums1 and nums2 at index 3 already satisfy the condition. There is no need for any swaps. So in this case, our count cnt is 0.

Finally, we compare results a and b from both cases. Since the result from the second case b (without any initial swap) is the minimum and is not -1, we add 1 to b because of the initial swap we considered. Therefore, b + 1 = 0 + 1 = 1. However, since we did not need that initial swap, the smallest number of operations needed to satisfy the conditions is actually just b, which is 0.

In conclusion, 0 swaps are required to make the last element of each array the maximum value within that respective array. We found that by not performing the swap on the last elements initially, our arrays were already in the desired state. Hence, the smallest number of operations needed is 0.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the code can be determined by looking at the two main components: the f function and the calls to this function within minOperations.

1. Function f: This function includes a for-loop that iterates over two arrays (nums1 and nums2). The loop runs for the length of the arrays minus one (n-1 times) because the [:-1] slice is used, which excludes the last element of each array. Inside the loop, there is a constant amount of work being done: a series of comparisons and a conditional increment of cnt. Therefore, the complexity of the f function is O(n-1) which simplifies to O(n).

2. Calls within minOperations: The function f is called twice inside minOperations. Since each call has a time complexity of O(n), and they are not nested, this does not change the overall time complexity of the code, which remains O(n).

Combining the parts, we maintain an overall time complexity of O(n) for the minOperations function.

Space Complexity

Analyzing the space complexity, there are no additional data structures that grow with the size of the input being used. The variables cnt, a, and b use a constant amount of space. The slices used in the zip function in f do not create new arrays; they just create views of the original arrays, so they don't add to the space complexity.

Therefore, the space complexity is O(1), as it does not depend on the size of the input arrays.

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