Problem Description

In this problem, we are given two strings s1 and s2. Both strings have the same length and contain only the characters 'x' and 'y'. Our objective is to make the two strings identical by performing swaps between the characters of the two strings. A swap involves taking one character from s1 and one character from s2 and exchanging them. The goal is to determine the minimum number of swaps required to make the two strings identical. If it's not possible to make the strings equal, we must return -1.

Intuition

The solution is based on counting how many characters are out of place when comparing both strings. We look for characters that are 'x' in s1 and 'y' in s2, and vice versa, since these are the only ones that need to be swapped to make the strings equal. There are two types of mismatches:

• An 'xy' mismatch: where s1 has an 'x' and s2 has a 'y' at the same position.
• A 'yx' mismatch: where s1 has a 'y' and s2 has an 'x' at the same position.

1. If we have an even number of 'xy' and 'yx' mismatches, we can always solve the problem.
2. We can swap odd mismatches within themselves, for instance, an odd count of 'xy' mismatches can be made even by performing one swap inside the 'xy' set (changing two 'xy' to two 'yx'). This adds one more swap to our answer for each odd count.
3. Each pair of mismatches ('xy' with 'yx') can be solved with one swap, so the pair count contributes directly to the total number of swaps.

The algorithm checks if there is an even total number of mismatches. If it's odd, we return -1 because we would always end up with one character that cannot be matched. If it's even, we sum the half of each count (because a swap fixes two mismatches) and case for odd counts, which contributes one additional swap for each.

Learn more about Greedy and Math patterns.

Solution Approach

The implementation of the solution uses a simple counter-based approach without any sophisticated data structures or patterns. The core of the solution revolves around counting the discrepancies between the two strings and categorizing them as xy or yx, followed by calculating the number of swaps needed based on these counts.

Let's walk through the steps present in the Python code provided:

1. Initialize two counters xy and yx to 0. These will hold the counts for mismatches where s1[i] is 'x' and s2[i] is 'y', and where s1[i] is 'y' and s2[i] is 'x', respectively.
2. Iterate over both strings in parallel by using the zip function, which allows us to obtain pairs of characters (a, b) from s1 and s2.
3. For each pair (a, b), increment the xy counter if a < b, which means we have an 'x' in s1 and a 'y' in s2. Similarly, increment the yx counter if a > b, which indicates a 'y' in s1 and an 'x' in s2.
4. Once we've counted all mismatches, we check whether the sum of xy and yx is even. If it's not ((xy + yx) % 2 is truthy), we return -1 because this means it's impossible to make the strings equal.
5. If the sum is even, we calculate the number of swaps. We calculate xy // 2 to see how many pairs of 'xy' mismatches can be swapped directly, which also applies to yx // 2 pairs of 'yx' mismatches. Additionally, for each type of mismatch, if there's an odd one out (checked by xy % 2 and yx % 2), it requires an extra swap.
6. The minimum number of swaps required is the sum of these values, which is xy // 2 + yx // 2 + xy % 2 + yx % 2, meaning the number of direct swaps for pairs plus the extra swaps for any remaining mismatch.

The logic applied here uses basic arithmetic and logic to deduce the number of required actions to align the two strings. It's a pragmatic approach that is both efficient and clear in its purpose.

Example Walkthrough

Let's illustrate the solution approach with a small example. Consider two strings s1 = "xyyx" and s2 = "yxyx".

Following the steps of the solution:

1. Initialize counters xy and yx to 0.

2. Iterate over both strings s1 and s2 and compare the corresponding characters using zip:

   • For the first pair (s1[0] = 'x', s2[0] = 'y'), since s1[0] < s2[0], increment xy to 1.
   • For the second pair (s1[1] = 'y', s2[1] = 'x'), s1[1] > s2[1], so increment yx to 1.
   • The third pair (s1[2] = 'y', s2[2] = 'y') matches, so no counter is incremented.
   • For the fourth pair (s1[3] = 'x', s2[3] = 'x'), again they match, so no counter is increased.

   After this step, xy equals 1, and yx equals 1.

3. Check if the sum of xy and yx is even: 1 + 1 = 2 which is even, so continue.

4. Calculate the minimum number of swaps:

   • Pairs of 'xy' mismatches that can be swapped directly: xy // 2 = 1 // 2 = 0.
   • Pairs of 'yx' mismatches that can be swapped directly: yx // 2 = 1 // 2 = 0.
   • An extra swap is needed for the one 'xy' mismatch: xy % 2 = 1 % 2 = 1.
   • An extra swap is needed for the one 'yx' mismatch: yx % 2 = 1 % 2 = 1.

5. The total number of swaps needed is: xy // 2 + yx // 2 + xy % 2 + yx % 2 = 0 + 0 + 1 + 1 = 2.

Therefore, for s1 = "xyyx" and s2 = "yxyx", it takes a minimum of 2 swaps to make the strings identical. The swaps that could be performed would be between the first character of s1 with the second character of s2, and the second character of s1 with the first character of s2. After these swaps, both s1 and s2 will be "xyxy".

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the given function is O(n), where n is the length of the strings s1 and s2. This is because the function consists of a single loop that iterates through all characters of s1 and s2 in a pairwise manner using the zip function. Within the loop, only constant time operations are performed (comparison and addition).

Space Complexity

The space complexity of the given function is O(1). There are only a fixed number of integer variables (xy and yx) initialized and used for counting the occurrences, which do not depend on the size of the input strings. Thus, the amount of additional memory required remains constant regardless of the input size.

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