Problem Description

The problem presents us with a string s. Our goal is to partition s into the maximum number of parts such that no letter appears in more than one part. This means that once a letter appears in a part, it cannot appear in any other part. After partitioning, when we concatenate all parts back together in the original order, the string should remain unchanged; it should be s.

The resulting output should be a list of integers, where each integer represents the length of one of these partitions.

Intuition

The intuition behind this solution is to first understand that a partition can only end when all instances of a particular character included in the partition have been encountered. The last occurrence of each character defines the maximum possible extent of a partition.

To solve the problem, we track the last occurrence of each character using a dictionary. The keys are the characters from the string s and the values are the indices of their last occurrences.

As we iterate over the string s:

1. We continually update a mx variable to keep track of the farthest index that we must reach before we can end a partition. This is updated by looking at the last occurrence of the current character.

2. When the current index i matches mx, it means we've included all occurrences of the characters seen so far in the current partition. At this point, we can end the current partition and start a new one.

3. We append the size of the current partition to the answer list. The size is calculated as the difference between the current index i and the start index of the partition j, plus one.

4. We then set the start of the next partition to be one index after the current index i.

In the end, we get a list of partition sizes that fulfill the requirements.

Learn more about Greedy and Two Pointers patterns.

Solution Approach

The solution uses a two-pass approach and utilizes a dictionary data structure to hold the last occurrence of each character in the string s.

1. Building a dictionary of last occurrences: The first pass is over the string s. As we iterate, we capture the last position or index of every unique character in the dictionary called last. This is constructed in the line last = {c: i for i, c in enumerate(s)}, making use of a dictionary comprehension.

2. Partitioning based on last occurrences: The next part of the solution is where the actual partitioning logic is implemented:

   • We initialize three variables: mx, j, and ans. The mx variable represents the furthest point a partition can reach before being cut. The j variable is the start index of the current partition, and ans is the list that will hold the sizes of the partitions.
   • We then iterate through the string s again, index by index, alongside the corresponding characters.
     • For each character at index i, we update mx to be the maximum value between the current mx and the last occurrence of the character c, like this: mx = max(mx, last[c]).
     • If the current index i equals mx, it means all characters in the current partition will not appear again in the following parts of the string. This is the right place to 'cut' this partition.
       • We calculate the size of the current partition by subtracting the start index j of this partition from the current index i and adding 1 (i - j + 1). This value is appended to the ans list.
       • Once we add the partition size to ans, we set the start j of the next partition to i + 1.

3. Output the result: After the iteration completes, we return the list ans which contains the sizes of all partitions.

The factors that make this algorithm work are:

• The dictionary captures the essential boundaries for our partitions by marking the last appearance of each character within the string.
• The two-pass mechanism ensures that we never 'cut' a partition too early, which ensures that each letter only appears in one part.
• The greedy approach - always updating mx to extend the partition as far as possible and only finalizing a partition when absolutely necessary - ensures that we maximize the number of partitions.

By following through with this approach, the function partitionLabels partitions the string optimally as per the problem statement.

Example Walkthrough

Let's consider a small example string s = "abac". We want to create partitions in such a way that no letters are repeated in any of the partitions and to maximize the number of partitions. The output should be a list of the lengths of these partitions.

1. Building a dictionary of last occurrences: The first pass over the string s will give us a dictionary last with the last position of every character.

   • For our example, last would be {'a': 2, 'b': 1, 'c': 3}.

2. Partitioning based on last occurrences:

   • We initialize mx to -1, j to 0, and ans to an empty list.
   • We then loop through the string s, with the indices and characters.
     1. At index 0, the character is 'a'. We update mx to be the maximum of mx and the last occurrence of 'a' (mx = max(-1, 2)), so mx becomes 2.
     2. At index 1, the character is 'b'. We update mx to be max(2, 1) as the last occurrence of 'b' is at index 1. Since 2 is greater, mx remains 2.
     3. At index 2, the character is 'a'. Since we have already accounted for the last occurrence of 'a', we don't need to update mx it is still the right end of our partition.
     4. At index 2 when i is equal to mx, we can 'cut' the partition here.
        • We append the size of this partition to ans, which is i - j + 1 or 2 - 0 + 1, so we append 3.
        • We also set j to the next index i + 1, which is 3, ready for the next potential partition.
     5. For the last character at index 3, 'c', the last occurrence is at 3 itself. We update mx to 3.
     6. Given that i equals mx at the last character, we have the end of another partition.
        • The size of the partition is again calculated as i - j + 1, yielding 3 - 3 + 1, so we append 1 to ans.

3. Output the result: After the iteration completes, ans contains the sizes [3, 1], denoting the lengths of the partitions aba and c. Hence, the final output is [3, 1].

By following the steps above, the input string abac has been partitioned into ["aba", "c"], maximizing the number of parts and ensuring no letter appears in more than one part. The lengths of the partitions are [3, 1] as per the expected output.

Solution Implementation

Time and Space Complexity

Time Complexity:

The time complexity for the provided code can be analyzed as follows:

• The first line within the partitionLabels function uses a dictionary comprehension to store the last appearance index of each character. This runs in O(n) time, where n is the length of string s, because it iterates through the entire string once.
• The subsequent for loop also runs through the string once and performs a constant number of operations within the loop. Hence, the complexity of this part is O(n).
• Overall, since both the creation of the dictionary and the for loop run in linear time relative to the size of the input s, and occur sequentially, the total time complexity of the function is O(n).

Space Complexity:

• The space complexity is influenced by the additional storage used:
  • The last dictionary which stores the index of the last occurrence of each character has at most O(min(n, 26)) space complexity because there are at most 26 letters in the alphabet. In the case where n < 26, the dictionary would have at most n elements.
  • The ans list where the sizes of the partitions are stored will have a space complexity of O(n) in the worst case, where each character is a partition on its own.
• Since the last dictionary has a fixed upper limit due to a finite alphabet, it's typically considered O(1) space, particularly in the context of large n.
• However, the list ans depends on the size of the input and can take up to O(n) space in the worst case scenario.
• Therefore, the overall space complexity of the function is O(n), reflecting the worst-case scenario where the output list could grow linearly with the input size.

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