Problem Description

The problem presents a scenario where you have a string text, and your task is to split it into k non-empty substrings such that each substring subtext_i equals the subtext at the position k - i + 1. This condition makes it so that the substrings are symmetrical around the center of text. The objective is to determine the largest possible value of k, which corresponds to the most such symmetrical substrings you can split text into. The problem requires you to take text and look for the longest sequences at the start and end that are identical, split them off, and then continue doing this iteratively until the entire text has been processed.

Intuition

The intuition behind the offered solution is to use a two-pointer approach. One pointer starts at the beginning of text (i) and the other at the end (j). The strategy is to look for the longest identical substring from these two starting points, which would be a candidate for subtext_1 and subtext_k. If a matching pair is found, then we can consider this as two parts of the decomposition, increment the number of substrings ans by 2, and move the pointers inward accordingly.

This process continues, searching for the next longest match, but now starting from the new position of i and j. This is done by expanding the length of the substring we're checking (k) until another match is found or we run out of characters to check. If at any point, it's impossible to find a matching pair for the current positions of i and j, it essentially means that the central part of text does not have a symmetrical counterpart, and we can increment ans by 1 to account for the remaining middle substring and then terminate the process.

The reason why this approach works is due to the greedy nature of the problem: slicing off the longest possible matching substrings at each step ensures that we're getting the most significant k value possible, as any longer substring would have naturally included shorter matching pairs that we find at each iteration.

Learn more about Greedy, Two Pointers and Dynamic Programming patterns.

Solution Approach

The solution implements a straightforward greedy algorithm using a two-pointer technique, which is a common pattern for exploring intervals or sequences within arrays and strings.

Initially, the count of substrings ans is set to zero. Two index pointers, i and j, are initialized to point to the start and end of the string text, respectively. The main loop continues as long as i is less than or equal to j, ensuring that we haven't completely processed all characters in text.

While in the loop, we initialize a variable k to 1, which represents the current length of the substring we're trying to match from both ends of the text. The variable ok is set to False, indicating whether we've found a matching pair of substrings. As long as the potential matching substrings don't overlap (ensured by checking that i + k - 1 < j - k + 1), we compare the substrings using slicing (i.e., text[i: i + k] == text[j - k + 1: j + 1]). If a match is found:

• We increment ans by 2 as we have found a pair of matching substrings.
• We update i and j to move past the matched substrings, effectively reducing the remaining portion of text that needs to be processed.
• We set ok to True since we've found a match.

If the inner while loop finishes without finding any matching substrings (ok remains False), it implies that the remaining text cannot be split symmetrically. In that case, we increment ans by 1 to account for the unique central substring and break the loop.

Finally, the function returns the number of decomposed substrings ans.

This solution relies on Python's ability to slice strings efficiently, and no additional data structures are used outside of the basic variables to keep track of indices and counts. The algorithm's efficiency primarily comes from the greedy approach of attempting to peel off the longest matching substrings first, thereby optimizing the overall number of splits, k.

Example Walkthrough

Let's illustrate the solution approach with a small example. Suppose we are given the string text = "abbaaccbbaa" and we want to split it into the maximum number of symmetrical substrings.

We initialize ans to 0, i to 0, and j to len(text) - 1 which is 10. Our text has a length of 11, where characters at positions 0 and 10 are 'a'.

The main loop starts, and we define k to be 1 inside the loop, and ok to be False. We check whether the substring at the start and end of text are equal, which they are (text[0:1] is 'a', and text[10:11] is also 'a'), hence we increment ans to 2, and update i to 1 and j to 9.

We again set k to 1. The next characters at the start and end (i = 1 and j = 9) of the remaining text are 'b' and 'b', which match. We continue this process and increment ans to 4, update i to 2 and j to 8.

At this point, text[i:i+1] is 'b' and text[j:j+1] is 'a', which do not match. We increase k to 2 and check again. Now, text[2:4] is 'ba' and text[7:9] is 'cb', so they also don't match. After increasing k to 3, we find that text[2:5] is 'baa' and text[6:9] is 'bba', that don't match either. We increment k until it's not possible to compare without overlap, it does not find any matching pairs, so it sets ok to False.

Since we have run out of characters to check for symmetrical substrings without overlap and ok is False, we increment ans by 1 to account for the remaining middle portion of the text and break out of the loop.

The central part of the text that doesn't have a symmetrical counterpart is cca. Therefore, we now have 5 symmetrical substrings within text: 'a', 'bb', 'cca', 'bb', 'a'.

The process terminates here and returns ans which is 5, corresponding to these substrings. This gives us the largest possible value of k for the given text, which is the most such symmetrical substrings we can split text into using the greedy algorithm approach described in the solution.

Solution Implementation

Time and Space Complexity

The time complexity of the code is given by the nested while loops. The outer loop runs for at most n/2 iterations, where n is the length of the input string text, because in each iteration at least one character is removed from each end of the text. The inner while loop could potentially run for n/2 iterations as well in the case where the matching substrings at the ends are just one character long, but located at the center of text. In the worst case scenario, every character is checked against its mirror character on the other side of the string, which happens n/2 times for half the length of the string. Therefore, the worst-case time complexity is O(n^2).

The space complexity of the code is O(1). This is because the algorithm uses a fixed amount of extra space regardless of the input size: variables ans, i, j, and k do not depend on the size of the input text.

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