Problem Description

In this problem, you're provided with a string s and an integer k. You are allowed to perform at most k operations on the string. In each operation, you may choose any character in the string and change it to any other uppercase English letter. The objective is to find the length of the longest substring (that is a sequence of consecutive characters from the string) that contains the same letter after you have performed zero or more, up to k, operations.

For example, given the string "AABABBA" and k = 1, you are allowed to change at most one character. The best way is to change the middle 'A' to 'B', resulting in the string "AABBBBA". The longest substring with identical letters is "BBBB", which is 4 characters long.

Intuition

The intuition behind the solution involves a common technique called the "sliding window". The core idea is to maintain a window (or a subrange) of the string and keep track of certain attributes within this window. As the window expands or contracts, you adjust these attributes and check to see if you can achieve a new maximum.

The solution keeps track of:

1. The frequency of each letter within the window: This is kept in an array called counter. Each index of this array corresponds to a letter of the English alphabet.

2. The count of the most frequent letter so far: During each step, the code calculates the current window's most frequent letter. This is stored in the variable maxCnt.

3. The indices i (end) and j (start) of the window.

The approach is as follows:

• Expand the window by moving i to the right, incrementing the counter of the current character.

• Update the maxCnt with the maximum frequency of the current character.

• Check if the current window size is greater than the sum of maxCnt (the most frequent character count) and k. If it is, then that means the current window cannot be made into a substring of all identical letters with at most k operations. If this happens, decrease the count of the leftmost character and move the start of the window to the right (increment j).

• Repeat this process until you have processed the entire string.

• Since the window size only increases or remains unchanged over time (because we only move i to the right and increment j when necessary), the final value of i - j when i has reached the end of the string will be the size of the largest window we were able to create where at most k operations would result in a substring of identical letters.

By the time the window has moved past all characters in s, you've considered every possible substring and the maximum viable window size is the length of the longest substring satisfying the condition. Hence the answer is i - j. This is an application of the two-pointer technique.

Learn more about Sliding Window patterns.

Solution Approach

To implement the solution, the approach capitalizes on several important concepts and data structures:

• Sliding Window Technique: This technique involves maintaining a window of elements and slides it over the data to consider different subsets of the data.

• Two Pointers: i and j are pointers used to represent the current window in the string, where i is the end of the window and j is the beginning.

• Array for Counting: An array counter of size 26 is used to keep count of all uppercase English letters within the current sliding window. Since there are only 26 uppercase English letters, it's efficient regarding both space and time complexity.

Here's a step-by-step of what happens in the code:

1. Initialization: Set up an array counter with length 26 to zero for all elements, representing the count of each letter. Initialize two pointers i and j to 0, and maxCnt to 0 which will store the maximum frequency of a single letter within the current window.

2. Sliding Window Expansion: Iterate over the string using the pointer i to expand the window to the right. For each character s[i], increment the count in the counter array at the position corresponding to the letter (found by ord(s[i]) - ord('A') where ord is a function that gets the ASCII value of a character).

3. Updating Maximum Letter Count: After updating counter for the new character, update maxCnt to reflect the maximum frequency of any single character in the current window.

4. Exceeding the Operation Limit: At each iteration, check if the current window size (i - j + 1) is greater than allowed (maxCnt + k). If it is, this means more than k replacements are required to make all characters in the current window the same. Therefore, you need to shrink the window by incrementing j, and decreasing the count of the character at the start of the window.

5. Continue Until the End: Keep repeating steps 2 to 4 until the end of the string is reached. At this point, since the window only grew or remained the same throughout the process, the difference i - j will be the length of the longest substring that can be achieved with at most k changes.

Using this approach, as you can see in the Python code, the function characterReplacement operates on the string efficiently by using a fixed amount of memory (the counter array) and makes a single pass over the string, thus the time complexity is O(n), where n is the length of the string.

Example Walkthrough

Let's walk through the solution approach using a small example: s = "ABAA" and k = 1.

1. Initialization: We start by initializing our counter array of size 26 to zero and the pointers i and j are both set to 0. maxCnt is also initialized to 0.

2. Sliding Window Expansion: Begin iterating through the string.

   • For the first character, 'A', counter[A] (consider counter[0] since 'A' is the first letter) is incremented to 1. Now maxCnt also becomes 1, as the only character in the window is 'A'.
   • Move i to the right to point to the second character s[1] which is 'B'. Increment counter[B] (consider counter[1] since 'B' is the second letter). maxCnt remains 1 because the frequency of both 'A' and 'B' is the same (1) in the window.

3. Updating Maximum Letter Count: As we continue, we update counter and maxCnt:

   • Increment i to point to s[2], which is 'A'. Now counter[A] is 2. We then update maxCnt to 2, as 'A' is now the most frequent letter within the window.

4. Exceeding the Operation Limit: Continue expanding the window by moving i to the right:

   • Increment i to point to s[3], which is 'A'. counter[A] is now incremented to 3. The window size is 4 (from s[0] to s[3]), and since maxCnt is 3 and k is 1, we satisfy the condition (window size) <= (maxCnt + k), which is 4 <= (3 + 1).

5. Shrinking the Window: At this point, since we're at the end of the string, we stop and observe that we did not have to shrink the window at any point. The largest window we could form went from index 0 to index 3 with one operation allowed to change a 'B' to an 'A', resulting in all 'A's.

The longest substring that can be formed from string "ABAA" by changing no more than 1 letter is "AAAA", which has a length of 4. So, our output is 4, which is the length from pointer j to i.

Using this step-by-step walkthrough, it is evident that this approach is both systematic and efficient in determining the length of the longest substring where at most k changes result in a uniform string.

Solution Implementation

Time and Space Complexity

The given code implements a sliding window algorithm to find the longest substring that can be created by replacing at most k characters in the input string s.

Time Complexity:

The time complexity of the code is O(n), where n is the length of the input string s. This is because:

• The algorithm uses two pointers i (end of the window) and j (start of the window) that move through the string only once.
• Inside the while loop, the algorithm performs a constant number of operations for each character in the string: updating the counter array, computing maxCnt, comparing window size with maxCnt + k, and incrementing or decrementing the pointers and counter.
• Although there is a max operation inside the loop which compares maxCnt with the count of the current character. This comparison takes constant time because maxCnt is only updated with values coming from a fixed-size array (the counter array with 26 elements representing the count of each uppercase letter in the English alphabet).
• No nested loops are dependent on the size of s, so the complexity is linear with the length of s.

Space Complexity:

The space complexity of the code is O(1):

• The counter array uses space for 26 integers, which is a constant size and does not depend on the length of the input string s.
• Only a fixed number of integer variables (i, j, maxCnt) are used, which also contributes to a constant amount of space.

In conclusion, the algorithm runs in linear time and uses a constant amount of additional space.

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