Problem Description

The problem is to find the substring that appears the most times in a given string s. The catch is that these substrings have to comply with two rules:

1. The quantity of unique characters in each substring must be less than or equal to maxLetters.
2. The length of the substring must be at least minSize and at most maxSize.

A key thing to note is that though substrings can have lengths anywhere from minSize to maxSize, we are interested in finding the maximum number of occurrences, and typically substrings with smaller lengths will have higher chances of repeated occurrence. The solution should return the maximum frequency of any such substring found in s that follows the above rules.

Intuition

When dissecting the problem, we focus on the constraints - unique characters within the substring and the substring's size. The fact that we're also looking for maximum occurrence leads to a strategy where smaller substrings are more likely to be repeated; hence, we prioritize evaluating substrings of minSize.

The intuition for the solution is to slide a window of size minSize across the string s and use a set to track the unique characters and a counter (hashmap) to count occurrences of each substring that fits the rules. For each window, we check the following:

1. If the set size is larger than maxLetters, we ignore this substring as it doesn't meet the first rule.
2. If the set size is within the limits, we add the substring to the counter and check if it's the most frequent one we've seen so far.

This way we iterate over the string once (O(n) time complexity, where n is the length of the string), and only consider substrings of length minSize because regardless of maxSize, any repeated instances of a longer substring will always contain a repeated minSize substring within it. Thus, minSize is the key to finding the maximum occurrence.

Learn more about Sliding Window patterns.

Solution Approach

The solution uses a sliding window algorithm and two data structures—a set and a counter from Python's collections module. The set is used to keep track of the unique characters within the current minSize window of the string, while the Counter is a hashmap to keep track of the frequency of each unique substring that fits the criteria.

Here are the steps in the implementation:

1. Initialize cnt as a Counter object to keep track of the frequency of each qualifying substring.
2. Initialize ans as an integer to keep track of the maximum frequency found.
3. Iterate over the string with a variable i ranging from 0 to len(s) - minSize + 1. This is your sliding window that will only consider substrings of size minSize.
4. At each iteration, create a substring t which is a slice of s starting from index i to i + minSize.
5. For the substring t, create a set ss to determine the number of unique characters it contains.
6. A crucial optimization here is that if len(ss) is greater than the maxLetters, the substring does not satisfy the required constraints, and no further action is taken for that window.
7. If len(ss) is less than or equal to maxLetters, add/subtract one to the cnt[t] for that substring sequence, which increases the count of how many times t has been seen.
8. Update ans which keeps track of the highest frequency seen so far by comparing it with the frequency count of t in cnt.
9. After the loop finishes, ans will hold the highest frequency of occurrence of any valid substring and is returned.

The algorithm sweeps through the string only once, making it efficient with time complexity of O(n) (for the loop) times O(k) (for the set creation, where k is at most minSize), and it only considers substrings of length minSize due to the provided insight that longer substrings will naturally contain these smaller, frequently occurring substrings if they are to be frequent themselves. The space complexity mainly depends on the number of unique substrings of length minSize that can be formed from s, which in worst cases is O(n).

Overall, the algorithm efficiently finds the most frequent substring of size within the given bounds that also contains no more than maxLetters unique characters.

Example Walkthrough

Let's consider a simple example with the following parameters:

• s: "ababab"
• maxLetters: 2
• minSize: 2
• maxSize: 3

Following the solution approach:

1. We initialize cnt as an empty Counter object and ans as 0.
2. We then iterate over s using a window size of minSize. In this example, minSize is 2, so we will be sliding through the string two characters at a time.
3. In the first iteration, i is 0, and our substring t is "ab" (from index 0 to 1). We create a set ss containing unique characters in "ab", which are {'a', 'b'}.
4. Since the size of ss is 2 and does not exceed maxLetters, which is also 2, we proceed to increment the count of this substring in cnt: cnt["ab"] += 1.
5. Now we slide the window by one and repeat. Our next substring is "ba" (from index 1 to 2), and ss will again be {'b', 'a'}. We increment cnt["ba"].
6. This process is repeated for the entire string: we then check "ab" again, and "ba" again. Each time, we are incrementing the count in cnt for the respective substring.
7. After we've processed each substring, we observe that "ab" and "ba" each have a count of 3 in cnt.
8. ans is then updated to be the maximum of the values in cnt, which in this case is 3.

The final answer, held by ans, is 3, indicating that the most frequent substrings of length minSize are "ab" and "ba," each appearing 3 times in the string s. Our example is now complete, demonstrating how the sliding window and counter work together to find the most frequent qualifying substring.

Solution Implementation

Time and Space Complexity

The time complexity of the code is O(n * minSize) where n is the length of the string s. This is because the code iterates over the string in slices of length minSize which takes O(minSize) time for each slice, and this is done for all starting points in the string, of which there are n - minSize + 1.

The space complexity of the code is O(n) in the worst case. This is due to the fact that the Counter could potentially store every unique substring of length minSize in the worst scenario where all possible substrings are unique. Since the length of each substring is bounded by minSize, this does not affect the order of space complexity.

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