Problem Description

You are given a string text which is indexed from 0. You are also given another string pattern, which is of length 2 and also indexed from 0. Both strings contain only lowercase English letters. You have the option to add one character either pattern[0] or pattern[1] to any position in the text string exactly once. This includes the possibility of adding the character at the start or the end of the text. Your task is to figure out the maximum number of times the pattern can be found as a subsequence in the new string after adding one character.

To clarify, a subsequence is a sequence that can be derived from another sequence by deleting some characters (or none) without altering the order of the remaining characters.

For example, given text = "ababc" and pattern = "ab", if we add "a" to the text to form "aababc", we can now see the subsequence "ab" occurs 4 times (indices 0-1, 0-3, 2-3, and 2-5).

Intuition

The intuition behind the solution lies in understanding what a subsequence is and how the addition of a character impacts the count of subsequences. The key insight is that by adding a character from the pattern to the text, we can increase the occurrences of that pattern as a subsequence.

We can track the occurrences of pattern[0] and the potential matches of the pattern as we iterate through the text. Each time we encounter the second character of the pattern in the text (pattern[1]), we know that any previous occurrences of pattern[0] could form a new subsequence match with it. We keep a cumulative count of pattern[0]'s occurrences as we scan through the text, adding to the answer whenever we see pattern[1].

Finally, we add the larger of the counts of pattern[0] and pattern[1] to ans because we can insert one additional character pattern[0] or pattern[1] to the text (at the best place possible), which will create the most additional subsequences.

This approach allows us to efficiently calculate the maximum number of subsequence counts possible by traversing the string only once.

Learn more about Greedy and Prefix Sum patterns.

Solution Approach

The solution uses a two-pass approach with a counter to keep track of occurrences of characters from the pattern.

1. Initialize a variable ans to keep track of the number of subsequences of pattern found in text.
2. Create a Counter object named cnt that will hold the frequency of each character we encounter in text.
3. Iterate over each character c in the text string.
   • If the current character c is the same as the second character in pattern (pattern[1]), we increment ans by the number of times the first character of pattern (pattern[0]) has appeared so far. This is because each occurrence of pattern[0] before pattern[1] could form a new valid subsequence with pattern[1].
   • Update cnt[c] by incrementing it by 1 to keep the count of each character.
4. After the iteration is complete, add to ans the maximum frequency between cnt[pattern[0]] and cnt[pattern[1]]. We can add one instance of either pattern[0] or pattern[1] anywhere in text to maximize the subsequence count. Adding the character from the pattern with the highest frequency will yield the highest number of subsequences.
5. Return the final answer ans, which represents the maximum number of times pattern can occur as a subsequence in the modified text.

Let's look at an example to better understand the approach:

• text = "ababc", pattern = "ab"
• As we iterate through text, we count occurrences of 'a' and 'b'.
• When we reach the first 'b' at index 1, ans is increased by the count of 'a' seen so far, which is 1 (cnt['a'] = 1).
• As we continue, every time we encounter 'b', we add the count of 'a's seen so far to ans.
• After the loop, we can add one more 'a' or 'b' (choosing 'a' is better in this case, as cnt['a'] > cnt['b']), adding cnt['a'] (which is 2) to ans.
• Finally, ans becomes 4, which is the maximum subsequence count after the addition.

The use of a Counter allows us to efficiently keep track of character frequencies, and the single pass approach with an additional step minimizes the time complexity, resulting in an elegant and effective solution.

Example Walkthrough

Let's apply the solution approach to a small example where text = "bbaca" and pattern = "ba".

1. We initialize ans = 0 because initially, we haven't found any subsequences of the pattern yet.
2. We create a Counter object cnt to keep track of occurrences of characters in text. It is initially empty.
3. We start iterating through the text:
   • We encounter the first 'b'. It's not the second character of pattern, so we just update cnt['b'] to 1.
   • We encounter the second 'b'. Again, it's not pattern[1], so we update cnt['b'] to 2.
   • We encounter 'a', and since it is pattern[0], we update cnt['a'] to 1 but do not alter ans since 'a' is not the second character in the pattern.
   • Next, we encounter 'c'. It's neither pattern[0] nor pattern[1], so we continue.
   • Finally, we encounter another 'a'. We update cnt['a'] to 2.
4. As we continue scanning, we find the last character 'b', which is pattern[1]. Now we add to ans the count of 'a' seen so far because 'a' followed by this 'b' can form another subsequence "ba". The ans is incremented by cnt['a'], which is 2. So now, ans = 2.
5. After the iteration is over, we look at the counts of 'b' and 'a' in cnt. We have cnt['b'] = 2 and cnt['a'] = 2. We can add one more character to text. To maximize the subsequences, we should choose to add the character with the maximum count.
6. In this case, both cnt['b'] and cnt['a'] are the same, so we can choose either. Let's choose to add another 'a'.
7. By adding an 'a', we will be able to create new subsequences "ba" with all existing 'b's. Therefore, we add cnt['b'] to ans, which results in an additional 2 subsequences.

So, the final answer ans is now 2 + 2 = 4, which represents the maximum number of times pattern = "ba" can occur as a subsequence in the modified text after adding one extra 'a'.

This walkthrough demonstrates the process of calculating the number of subsequences of a given pattern in a text by iteratively counting characters, leveraging the subsequence definition, and maximizing the outcome by intelligently adding a character to the text based on the counts obtained.

Solution Implementation

Time and Space Complexity

The given Python code calculates the number of times a certain pattern of two characters can be inserted into a text such that the subsequence count of that pattern is maximized.

Time Complexity

The time complexity of the function is determined by a single pass over the text string text, which has length n. During this pass, for each character c in text, the code updates the Counter object cnt and in some cases increments the answer ans.

Accessing and updating the count in Counter for each character is generally O(1) complexity assuming a good hash function (as Counter is a type of dict in Python, and dictionary access is generally considered constant time).

The increment of ans based on cnt[pattern[0]] also occurs in constant time.

Therefore, because we have a single pass over the text with constant-time operations within the loop, the overall time complexity of the function is O(n), where n is the length of the input string text.

Space Complexity

The space complexity is determined by the additional space used which is mainly the Counter object cnt. In the worst case, cnt could store a count for every unique character in text. The number of unique characters in text could be up to the size of the character set but is typically much smaller.

If we only consider the length of text, the space complexity would be O(u) where u is the number of unique characters in the text, which is less than or equal to n.

However, since the space taken up by cnt does not scale with the size of the input text, but rather with the number of unique characters, it might also be fair to consider it O(1) space, under the assumption that the character set size is fixed and not very large (e.g., ASCII characters).

So, the space complexity is either O(u) or O(1), depending on whether you count the fixed size of the character set as constant or not.

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