Problem Description

The problem presents a scenario where a teacher is attempting to create a true/false test that is deliberately designed to be confusing for the students. The confusion is maximized by having the longest possible sequence of consecutive questions with the same answer, either true ('T') or false ('F').

The answerKey string is given, representing the correct answers for each question, where each character is either 'T' for true or 'F' for false. Additionally, you are given an integer k which indicates the maximum number of times the teacher can change any answer in the answerKey from 'T' to 'F' or vice versa.

The objective is to find the maximum length of a subsequence of consecutive characters in the answerKey that can be the same character after performing at most k modifications.

Intuition

The intuition behind the solution relies on using a sliding window approach. The sliding window is a technique that can efficiently find a subarray or substring matching a certain condition within a larger array or string. In the context of this problem, the sliding window represents the sequence of consecutive answers that can be made the same (all 'T's or all 'F's) with up to k changes.

The main idea is to maintain two pointers, l and r, which respectively represent the left and right ends of the sliding window. These pointers move along the answerKey, expanding to the right (r increases) as long as there are characters not matching the desired one and the number of allowed changes k has not been exceeded. If the count of non-matching characters within the window exceeds k, the window is contracted by moving the left pointer (l) forward.

By performing this process twice - once to maximize the number of 'T's and once for 'F's - and taking the maximum of these two values, the solution finds the longest sequence of consecutive identical answers that can be obtained through at most k modifications.

The function get(c, k) handles this sliding window process by taking a character c ('T' or 'F') and the number of allowed changes k. It keeps track of the window by incrementing pointers and modifying the allowed changes k accordingly. The length of the longest valid window found while scanning with either character is then returned.

Finally, the maxConsecutiveAnswers function simply returns the maximum length found between the two separate iterations of the get function with 'T' and 'F' as the target characters.

Learn more about Binary Search, Prefix Sum and Sliding Window patterns.

Solution Approach

The solution uses a sliding window algorithm to determine the maximum number of consecutive 'T's or 'F's that can be obtained in the answerKey, given that we can change up to k answers. The definition of the function get(c, k) inside the Solution class plays a crucial role in implementing this algorithm.

The variables l and r are pointers used to denote the left and right bounds of the sliding window, initially set to -1 since the window starts outside the bounds of the answerKey string. The while loop inside get(c, k) drives the expansion of the window to the right.

For every iteration, r is incremented to include the next character in the window. If this character does not match the target character c, k is decremented, which reflects that we use one operation to change this character to c.

If at any point the number of operations used (k) becomes negative, this means the current window size cannot be achieved as it would require more than k operations to make all characters the same as c. To rectify this, the left bound of the window (l) is moved to the right (l += 1), effectively shrinking the window from the left side. If the character at the new left bound (answerKey[l]) was a mismatch and we previously used an operation to adjust it, k is incremented since that operation is no longer needed for the new smaller window.

After expanding the window as much as possible without exceeding the maximum allowed modifications (k), the function calculates the window's size (r - l) and returns this value to the caller.

The solution approach calls get(c, k) twice, once with c = 'T' and once with c = 'F', to compute the maximum length of consecutive answers that can be obtained for both 'T' and 'F', respectively. The maximum of these two values is returned by the maxConsecutiveAnswers as the final answer to the problem.

Essentially, the algorithm leverages the sliding window technique to iterate over answerKey while keeping the modifications within the limit imposed by k, and dynamically adjusting window bounds to maximize the size of the window which represents a sequence of identical characters.

By returning the maximum between one pass with 'T' as the target character and one pass with 'F', the algorithm ensures that it considers both scenarios—maximizing consecutive 'T's and 'F's—ultimately returning the best possible result.

This solution is efficient as both pointers (left and right) move across the answerKey in one direction only, keeping the time complexity linear, O(n), where n is the length of the answerKey.

Example Walkthrough

Let's consider a small example to illustrate the solution approach.

Suppose we have: answerKey = "TFFTFF" and k = 1.

The aim is to find the maximum length of consecutive answers after changing at most one answer ('T' to 'F' or vice versa).

Step-by-Step Process to Find Maximum Consecutive 'T's

Here, we'll go through the process using get('T', k):

1. Initialize pointers l and r to -1 so that our sliding window is initially empty.
2. Start scanning the answerKey from left to right.
3. The r pointer moves right one step. When r = 0, the character is 'T', which matches our target, so we don't need to use a change.
4. Move r to 1, the character is 'F', which is not our target, so we decrement k by 1 (now k = 0 because we need one change to make it a 'T').
5. We can't use more changes (since k = 0); continue to move r to the right.
6. At r = 2, we have another 'T', no changes needed.
7. At r = 3, we have 'F'. Since we can't change it (k is already 0), we need to move l up to release a change. Move l to 0 and since answerKey[l] was a 'T', no k increment.
8. Now we can move r forward. At r = 3, we change 'F' to 'T'. (Now k = -1, which is not allowed, so we must adjust l again).
9. Move l to 1. Now k becomes 0 again because we've released the change at r = 1.
10. At r = 4 and 5, we have 'F's, but since we have no changes left, we can't include these in our sequence.

The largest sequence of 'T's that we can obtain is from l + 1 to r, which is from index 2 to 3 (subsequence "TT"). Its length is 3 - 1 = 2.

Step-by-Step Process to Find Maximum Consecutive 'F's

Repeat the same method using get('F', k):

1. Initialize l and r to -1.
2. Scan answerKey from left to right, modifying at most k characters to 'F'.
3. The process will find that the longest sequence of 'F's, after one modification, can be from index 1 to 5 (subsequence "FFFFF") if we change the 'T' at index 2 to an 'F'.
4. Its length is 5 - 0 = 5.

Comparing both cases:

• Maximum length for 'T's: 2
• Maximum length for 'F's: 5

Final Answer

The final answer is the maximum of these two lengths, which in this example is 5 since the sequence of 'F's is longer.

This small illustration explains how the sliding window algorithm is used to determine the longest possible sequence of consecutive 'T's or 'F's that can be achieved by making at most k modifications. By trying both cases separately and returning the maximum value obtained from them, the algorithm yields the optimal solution to the problem.

Solution Implementation

Time and Space Complexity

Time Complexity

The function get uses a sliding window approach to count the maximum consecutive answers by flipping a certain number of T or F. The inner while loop runs once for each character in answerKey as both l and r iterate over the indices from start to end without stepping backwards. Thus, the complexity of the function get is linear with respect to the length of the answerKey, which is denoted as n. Since the function get is called twice (once for T and once for F), the overall time complexity of maxConsecutiveAnswers is O(n), as the two linear passes are additive, not multiplicative.

Space Complexity

The space complexity of the code is O(1). The function get uses only a fixed number of variables (l, r, and k), which doesn't depend on the size of the input answerKey. There are no data structures used that scale with the input size, and the memory usage remains constant regardless of the length of answerKey.

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