Problem Description

In this problem, we are given a string s. Our task is to determine the number of ways we can split this string into two non-empty substrings s_left and s_right such that their concatenation adds back to the original string s (i.e., s_left + s_right = s) and the number of unique characters in s_left is the same as the number of unique characters in s_right. A split that satisfies these conditions is called a good split. We need to return the total count of such good splits.

Intuition

To arrive at the solution, we can use a two-pointer technique that counts the number of unique characters in the left and right parts of the string incrementally. We can start by counting the distinct letters in the entire string s and create a set to keep track of the distinct letters we have seen so far as we iterate through the string from left to right.

For each character c in the string s, we perform the following steps:

• We add the character c to the set of visited (or seen) characters, which represents the left part of the split (s_left).
• We decrement the count of c in the total character count, which essentially represents the right part of the split (s_right).
• If the count of c after decrementing becomes zero, it means that there are no more occurrences of c in the right part (s_right), and we can remove c from the character count for the right part.
• After each character is processed, we check if the size of the visited set (number of unique characters in s_left) is the same as the number of characters remaining in s_right. If they are equal, we have found a good split, and we increment our answer (ans) by one.

By the end of this process, ans will hold the total number of good splits that can be made in string s.

Learn more about Dynamic Programming patterns.

Solution Approach

The implementation of the solution follows these steps:

1. Initialize a Counter object from Python's collections module for string s. This Counter object will hold the count of each character in the string, which we’ll use to keep track of characters in the right part of the split (s_right).

2. Create an empty set named vis to track the distinct characters we have encountered so far, which represents the left part of the split (s_left).

3. Set an answer variable ans to zero. This variable will count the number of good splits.

4. Iterate through each character c in the string s:

   • Add the current character c to the vis set, indicating that the character is part of the current s_left.
   • Decrement the count of character c in the Counter object, reflecting that one less of the character c is left for s_right.
   • If the updated count of character c in the Counter becomes zero (meaning c no longer exists in s_right), remove c from the Counter to keep the counts and distinct elements accurate for remaining s_right.
   • Evaluate if there is a good split by comparing the size of the vis set with the number of remaining distinct characters in s_right as denoted by the size of the Counter. If they are the same, it means we have an equal number of distinct characters in s_left and s_right, and thus, increment ans by one.

5. After the for loop completes, return the value of ans.

Throughout this process, we are using a set to keep track of the unique characters we've seen which is an efficient way to ensure we only count distinct letters. Utilizing a Counter allows us to accurately track the frequency of characters as we 'move' characters from right to left by iterating through the string, effectively keeping a live count of what remains on each side of the split. Comparing the lengths of the set and the Counter keys at each step allows us to check if a good split has been achieved without needing to recount characters each time.

The solution is efficient because it only requires a single pass through the string s, which makes the time complexity of this approach O(n), where n is the length of the string.

Here is the implementation encapsulated in the class Solution:

from collections import Counter

class Solution:
    def numSplits(self, s: str) -> int:
        cnt = Counter(s)  # Initial count of all characters in `s`
        vis = set()       # Set to keep track of unique characters seen in `s_left`
        ans = 0           # Counter for number of good splits

        # Looping through every character in the string
        for c in s:
            vis.add(c)
            cnt[c] -= 1
            if cnt[c] == 0:
                cnt.pop(c)
            ans += len(vis) == len(cnt)
        return ans

Example Walkthrough

Imagine the string s is "aacaba". We need to calculate the number of good splits for this string.

1. First, we create a counter from the whole string s which will give us {'a': 4, 'c': 1, 'b': 1} showing the counts of each character in s.

2. We then initialize the set vis to keep track of the unique characters seen in s_left (initially empty) and set our answer count ans to 0.

3. As we iterate through the string:

   • For the first character 'a', we add it to vis (now vis is {'a'}) and decrement its count in cnt (now cnt is {'a': 3, 'c': 1, 'b': 1}). The lengths of vis and cnt are not equal, so ans remains 0.

   • Moving to the second character 'a', we add it to vis (which remains {'a'} since 'a' is already included) and decrement its count in cnt (now cnt is {'a': 2, 'c': 1, 'b': 1}). The lengths of vis and cnt are still not equal, so ans remains 0.

   • Now we come to the third character, 'c'. We add 'c' to vis (now vis is {'a', 'c'}) and decrement its count in cnt (now cnt is {'a': 2, 'c': 0, 'b': 1}), and since the count of 'c' has reached 0, we remove 'c' from cnt (now cnt is {'a': 2, 'b': 1}). The lengths of vis and cnt are now equal (2 each), so we increment ans to 1.

   • Then we process the fourth character 'a'. After adding 'a' to vis (which remains {'a', 'c'}) and decrementing its count in cnt (now cnt is {'a': 1, 'b': 1}), we find that the lengths of vis and cnt are still equal (2 each), so we increment ans to 2.

   • The fifth character is 'b'. We add 'b' to vis (now vis is {'a', 'c', 'b'}) and decrement its count in cnt (now cnt is {'a': 1, 'b': 0}), and remove 'b' from cnt since its count is now 0 (now cnt is {'a': 1}). Lengths of vis and cnt are not equal, so ans remains 2.

   • Finally, we process the last character 'a'. We add 'a' to vis (which remains {'a', 'c', 'b'}) and decrement its count in cnt (now cnt is {'a': 0}), and then remove 'a' from cnt since its count is 0 (now cnt is empty). The lengths of vis and cnt are not equal, so ans remains 2.

4. After the loop finishes, since there were two points where the count of unique characters in s_left and s_right were the same, the value of ans is 2.

Therefore, the total number of good splits for the string "aacaba" is 2.

Solution Implementation

Time and Space Complexity

Time Complexity

The given code snippet involves iterating over each character of the string s precisely once. Within this single iteration, the operations performed involve adding elements to a set, updating a counter (a dictionary under the hood), checking for equality of lengths, and incrementing an answer counter.

• Adding elements to the vis set has an average case time complexity of O(1) per operation.
• Updating the counts in the Counter and checking if a count is zero is also O(1) on average for each character because dictionary operations have an average case time complexity of O(1).
• The equality check len(vis) == len(cnt) is O(1) because the lengths can be compared directly without traversing the structures.

Thus, we have an average case time complexity of O(n), where n is the length of the string s.

Space Complexity

The space complexity is determined by the additional data structures used:

• A Counter object to store the frequency of each character in s. In the worst case, if all characters in s are unique, the counter would hold n key-value pairs.
• A set object to keep track of the characters that we have seen as we iterate. This could also hold up to n unique characters in the worst case.

Both the Counter and the set will have a space complexity of O(n) in the worst case. Therefore, the overall space complexity is O(n).

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