Problem Description

The task is to create a "good" string by removing characters from the given string s. A "good" string is defined as one in which no two distinct characters have the same number of occurrences (or frequency). The goal is to find and return the minimum number of character deletions required to achieve a "good" string.

Frequency is simply how many times a character appears in the string. As an example, in the string aab, the character a has a frequency of 2, while b has a frequency of 1. The varying frequencies of each character play a crucial role in determining what constitutes a "good" string.

Intuition

To solve this problem, we need to adjust the frequency of characters so that no two characters have the same count. To minimize the number of deletions, we should try to decrease the frequencies of the more common characters as opposed to the less common ones, since generally this will lead to fewer total deletions.

The solution approach involves these steps:

1. First, we count the frequency of each character in the string using a counter data structure.
2. Then, we sort the frequencies in descending order so we can address the highest frequencies first.
3. We initialize a variable pre to inf which represents the previously encountered frequency that has been ensured to be unique by performing the necessary deletions.
4. We iterate over each sorted frequency value:
   • If pre is 0, indicating that we can't have any more characters without a frequency, we must add all of the current frequency to the deletion count since having a frequency of 0 for all subsequent characters is the only way to ensure uniqueness.
   • If the current frequency is greater than or equal to pre, we must delete enough of this character to make its frequency one less than pre and update pre to be this new value.
   • If the current frequency is less than pre, we simply update pre to this frequency as no deletions are needed, it's already unique.

Throughout this process, we keep track of the total number of deletions we had to perform. Our goal is to maintain the property of having unique frequencies as we consider each frequency from high to low. By considering frequencies in descending order, we ensure that we are minimizing the number of deletions needed by prioritizing making more frequent characters less frequent.

In the solution code, the Counter from the collections module is used to count the frequencies, sorted() gets the frequencies in descending order, and a for loop is used to apply the described logic, updating the ans variable to store the total number of deletions required. inf is used as a placeholder for comparison in the loop to handle the highest frequency case on the first iteration.

Learn more about Greedy and Sorting patterns.

Solution Approach

The solution involves implementing a greedy algorithm which operates with the data structure of a counter to count letter frequencies and a sorted list to process those frequencies. The pattern used here is to always delete characters from the most frequent down to the least, ensuring no two characters have the same frequency. Here's how the implementation unfolds:

1. Count Frequencies: The Counter from Python's collections module is used to create a frequency map for each character in the string. The Counter(s) invocation creates a dictionary-like object where keys are the characters, and values are the count of those characters.

2. Sort Frequencies: These frequency values are then extracted and sorted in descending order: sorted(cnt.values(), reverse=True). The sorting ensures that we process characters by starting from the highest frequency.

3. Initialize Deletion Counter and Previous Frequency: An integer ans is initialized to count the deletions needed and pre is set to inf to ensure that on the first iteration the condition v >= pre will be false.

4. Process Frequencies:

   • Iterate over the sorted frequency list.
   • If pre has been decremented to 0, it means we can no longer have characters with non-zero frequency (as we cannot have frequencies less than 0). Thus, for the current frequency v, all characters must be deleted, hence ans += v.
   • If the current frequency v is greater than or equal to pre, we decrement v to one less than pre to maintain frequency uniqueness, which makes pre the new current frequency minus 1, and increment ans by the number of deletions made, v - pre + 1.
   • If v is less than pre, it's already unique, so update pre to v and continue to the next iteration.

5. Return Deletions: After processing all character frequencies, the sum of deletions stored in ans is returned, which is the minimum number of deletions required to make the string s "good".

In terms of complexity, the most time-consuming operation is sorting the frequencies, which takes O(n log n) time. Counting frequencies and the final iteration take linear time, O(n), making the overall time complexity O(n log n) due to the sort. The space complexity is O(n) for storing the character frequencies and the sorted list of frequencies.

By implementing this greedy approach, we ensure that the process is efficient and that the least number of deletions are performed to reach a "good" string.

Example Walkthrough

Let's go through a small example to illustrate the solution approach. Suppose we have the string s = "aabbccddd". We want to create a "good" string by removing characters so that no two characters have the same frequency. Let's apply the steps outlined in the solution approach:

1. Count Frequencies: First, we use a Counter to get the frequencies of each character in s. The counter reveals the frequencies as follows: {'a': 2, 'b': 2, 'c': 2, 'd': 3}.

2. Sort Frequencies: We sort these values in descending order, which gives us [3, 2, 2, 2].

3. Initialize Deletion Counter and Previous Frequency: We initialize ans = 0 for counting deletions and pre = inf as the previous frequency.

4. Process Frequencies: Now, we iterate over the sorted list and apply the logic:

   • For the first frequency 3, since pre is inf, we don't need to delete anything. We update pre to 3.
   • Next, we look at the frequency 2. Since pre is 3, we can keep it as is and update pre to 2.
   • For the next frequency 2, it's equal to pre, so we need to delete one character to make it 1 (one less than the current pre). We increment ans by 1 and update pre to 1.
   • For the last frequency 2, we again need to make it less than pre, so we delete two characters this time, making it 0. We increment ans by 2 and since pre is already 1, we note that we can't reduce it further and any additional characters would need to be fully deleted.

5. Return Deletions: We've finished processing and have made 3 deletions in total (ans = 3). The result is that the minimum number of deletions required to make s a "good" string is 3.

After these steps, the initial string aabbccddd has been transformed into a "good" string aabbcd by deleting two 'd's and one 'c'. Each remaining character (a, b, c, d) has a unique frequency (2, 2, 1, 1 respectively).

And following the time complexity analysis, most of our time expense was in the sorting step, with the rest of the process performed in linear time relative to the length of the string.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the code mainly consists of three parts:

1. Counting the frequency of each character in the string s which takes O(n) time, where n is the length of the string s.

2. Sorting the counts which take O(k log k) time, where k is the number of unique characters in the string s. In the worst case, k can be up to n if all characters are unique.

3. Iterating over the sorted counts to determine the minimum number of deletions which takes O(k) time.

Thus, the overall time complexity is O(n + k log k + k), which simplifies to O(n + k log k) because n is at least as large as k.

Space Complexity

The space complexity of the code mainly comes from two parts:

1. Storing the character counts which require O(k) space, where k is the number of unique characters in s.

2. The sorted list of counts which also requires O(k) space.

Thus, the space complexity is O(k).

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