Problem Description

In this problem, we are given two zero-indexed (meaning indexing starts from 0) strings word1 and word2. We are allowed to perform a move which involves choosing two indices i from word1 and j from word2 (both indices must be within the bounds of their respective strings) and swapping the characters at these positions, i.e., word1[i] and word2[j].

The objective is to determine if it is possible to equalize the number of distinct characters in both word1 and word2 using exactly one such move. If this is possible, our function should return true. Otherwise, it should return false.

Intuition

The intuition behind the solution is based on counting characters in each string and then considering every possible swap to see if it can lead us to the goal of equalizing the distinct number of characters in both strings with just one move.

To keep track of the characters, we use two arrays cnt1 and cnt2 of size 26 (assuming the input strings consist of lowercase alphabetic characters only). Each array corresponds to counting occurrences of characters in word1 and word2, respectively.

Here's the thinking process to arrive at the solution:

• We first count how many times each character appears in both word1 and word2.
• Then, we iterate through the counts of both cnt1 and cnt2. For every pair of characters (i, j) where cnt1[i] and cnt2[j] are both non-zero (indicating that character i is available to swap in word1 and character j in word2), we emulate a swap.
• After the virtual swap, we calculate the number of distinct characters currently present in cnt1 and cnt2.
• We check if the number of distinct characters is now equal in both. If it is, we return true.
• If it isn't, we revert the swap back to its original state and continue with the next possible swap pair.

The reason we try every possible swap is that the distinct number of characters is influenced by both the characters being swapped in and out. So, to find out if any swap can achieve our goal, we have to examine each possibility.

Solution Approach

The implementation of the solution follows a simple yet efficient brute-force approach to verify whether a single swap can make the number of distinct characters equal in both strings. Here's a walk-through:

• Data Structures: Two arrays cnt1 and cnt2 are utilized, each with a length of 26 corresponding to the 26 letters in the English alphabet. These arrays are used to count the occurrences of each character in word1 and word2 respectively.

• Counting Characters: Iterate over word1 and word2 to count each character. This is done by converting the character to its ASCII value with ord(c), subtracting the ASCII value of 'a' to normalize the index to 0-25, and incrementing the count at that index in cnt1 or cnt2. This looks like the following:

  for c in word1:
      cnt1[ord(c) - ord('a')] += 1
  for c in word2:
      cnt2[ord(c) - ord('a')] += 1

• Attempting Swaps: The next step is to consider every possible swap. For this, nested loops are used to go over every index i in cnt1 and every index j in cnt2. If cnt1[i] and cnt2[j] are both greater than 0 (meaning both characters are present in their respective strings), a virtual swap is performed:

  cnt1[i], cnt2[j] = cnt1[i] - 1, cnt2[j] - 1
  cnt1[j], cnt2[i] = cnt1[j] + 1, cnt2[i] + 1

  This emulates moving one occurrence of the i-th character from word1 to word2 and one occurrence of the j-th character from word2 to word1.

• Checking Distinct Characters: After emulating the swap, a check is performed to determine if the number of distinct characters in both arrays is the same. This is done by summing up the count of indices greater than 0 in both arrays:

  if sum(v > 0 for v in cnt1) == sum(v > 0 for v in cnt2):
      return True

• Reverting Swap: If the numbers are not equal, the swap is reverted:

  cnt1[i], cnt2[j] = cnt1[i] + 1, cnt2[j] + 1
  cnt1[j], cnt2[i] = cnt1[j] - 1, cnt2[i] - 1

  And the algorithm continues to the next possible swap.

• Result: If a successful swap that balances the number of distinct characters is found, the function returns True. If no such swap is found after all possibilities have been considered, the function eventually returns False.

This algorithm is efficient in the sense that it goes through a limited set of possible swaps (at most 26 * 26) and requires no extra space besides the two counting arrays.

Example Walkthrough

Let's illustrate the solution approach with a small example:

Suppose we have word1 = "abc" and word2 = "def". Our goal is to determine if we can equalize the number of distinct characters in both words by performing exactly one swap.

First, we initialize two arrays to count the occurrences of each character, cnt1 and cnt2. After iterating through each word:

cnt1 (for word1) would be [1, 1, 1, 0, 0, 0, ..., 0] corresponding to [a, b, c, ..., z]. cnt2 (for word2) would be [0, 0, 0, 1, 1, 1, ..., 0] also corresponding to [a, b, c, ..., z].

Now, we attempt every possible swap between characters in word1 and word2:

1. Trying to swap a from word1 with d from word2

We adjust our count arrays to reflect this potential swap:

• cnt1 becomes [0, 1, 1, 1, 0, 0, ..., 0]
• cnt2 becomes [1, 0, 0, 0, 1, 1, ..., 0]

We now check if the number of distinct characters in both is the same:

• cnt1 has 3 distinct characters (b, c, d)
• cnt2 has 3 distinct characters (a, e, f)

Since the number of distinct characters is equal, we return True.

However, if we need to continue to illustrate the rest of the process, we would revert this virtual swap and try other possibilities:

Revert back to original counts:

• cnt1 becomes [1, 1, 1, 0, 0, 0, ..., 0]
• cnt2 becomes [0, 0, 0, 1, 1, 1, ..., 0]

2. Trying to swap a from word1 with e from word2

... and so on for every character in word1 vs every character in word2.

For the sake of this example, we already found a swap that works, so there's no need to continue. The function would now return True. If a swap could not equalize the number of distinct characters, then we would finally return False after exhausting all the combinations.

Solution Implementation

Time and Space Complexity

Time Complexity

The provided code iterates over each character in word1 and word2, counting the frequency of every character. Then it has nested loops where it iterates over the counts of characters in cnt1 and cnt2 arrays (size 26 for each letter of the alphabet) and performs operations to check if swapping characters could make the frequency of non-zero characters in the count arrays equal.

The time complexity of counting characters is O(n) for each word, where n is the length of the word. However, the nested loops create a bigger time complexity issue. There are 26 possible characters, leading to 26*26 comparisons in the worst case which, is O(26^2) or simply O(1) since 26 is a constant and does not change with the input size.

Combining these, the overall time complexity is primarily affected by the character counting, so O(n) for word1 plus O(m) for word2, where n and m are the lengths of word1 and word2 respectively. Since the 26*26 operations are constant time and do not scale with n or m, the insignificant additional constant time doesn't affect the overall complexity.

The total time complexity is O(n + m), where n is the length of word1 and m is the length of word2.

Space Complexity

The space complexity is much simpler to analyze. The space required by the algorithm is the space for the two count arrays cnt1 and cnt2 which hold the frequency of each character. As these arrays have a fixed size of 26, regardless of the input size, the space complexity is O(1).

Putting it all together, the space complexity is O(1) because it only requires fixed space for the frequency counts and a few variables for iteration and comparison, which does not scale with the input size.

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