1745. Palindrome Partitioning IV

Problem Description

The problem requires us to determine whether a given string s can be divided into three non-empty substrings that each form a palindrome. A palindrome is a string that reads the same forward and backward—for example, racecar or madam. If it is possible to split the input string into such three palindromic segments, we return true; otherwise, we return false.

Intuition

The solution approach involves first understanding the characteristics of a palindrome. A string is a palindrome if the first and last characters are the same, and if the substring between them (if any) is also a palindrome. The intuition behind the solution involves checking all possible ways of splitting the string into three substrings and testing for each split if all three resulting substrings are palindromes.

To decide whether a substring is a palindrome, we can precompute the palindromic status of all possible substrings. We create a 2D array g where g[i][j] is True if the substring from index i to index j is a palindrome. This precomputation uses dynamic programming. Once we have this 2D array, we can iterate through all possible splits with two nested loops, using the g array to check if each potential three-segment split consists of palindromes. If such a three-palindrome partitioning exists, we return True; if we finish our iterations without finding one, we return False.

The dynamic programming part to fill the g array checks palindromes by comparing the characters at the indices i and j, and by ensuring that the substring between them is a palindrome (by checking the entry g[i+1][j-1]).

By precomputing the palindrome statuses, we save time in the later stage where we're iterating over potential partition indices, allowing this solution to be efficient and effective for the problem.

Learn more about Dynamic Programming patterns.

Solution Approach

The solution employs dynamic programming and precomputation to determine the palindromic nature of all substrings in the string s. The data structure used for this is a 2-dimensional array g where g[i][j] holds a boolean value indicating whether the substring starting at index i and ending at index j is a palindrome.

Here's a step-by-step breakdown of the implementation:

1. Initialize a 2D array g of size n * n filled with True, where n is the length of the string s. Since a substring of length 1 is trivially a palindrome, the diagonal of the array g (where i==j) represents these substrings and is thus initialized to True.

2. Populate the g array using a nested loop. Starting from the end of the string, we move towards the beginning (decreasing i). For each i, we iterate j from i+1 to the end of the string. We update g[i][j] to True only if the characters at i and j are equal and the substring g[i+1][j-1] is a palindrome. This check ensures that we confirm the palindromic property for a substring by comparing its outer characters and using the previously computed statuses of its inner substrings.

3. Once the g array is computed, we use two nested loops to try all possible partitions. The first loop iterates over the first potential cut in the string (denoted by i; it marks the end of the first substring), which goes from index 0 to n-3. The second loop goes over the second potential cut (denoted by j; it marks the end of the second substring), which starts one position after i and goes until n-2. These loops make sure that each of the three resulting substrings is non-empty.

4. For each pair of positions (i, j), we check if the substrings s[0...i], s[i+1...j], and s[j+1...end] (end being n-1, inclusive) are palindromes. This is done using the g array: if g[0][i], g[i+1][j], and g[j+1][n-1] are all True, this means that the current partition forms three palindromic substrings.

5. If a valid partition is found, we immediately return True.

6. If no valid partition is found after going through all possible pairs of (i, j), then it's not possible to split the string into three palindromic substrings, and we return False.

The use of dynamic programming makes this process efficient since we avoid recomputing the palindrome status of substrings when checking possible partition positions.

Example Walkthrough

Let's use the string "abacaba" to illustrate the solution approach.

1. We first initialize the 2D array g of size 7 x 7 (since "abacaba" has 7 characters) with True on the diagonal, as any single character is a palindrome by itself.

2. We then fill the g array with the palindromic status for all substrings. For example, g[0][1] will be False because s[0] is 'a' and s[1] is 'b', and "ab" is not a palindrome. We continue this for all possible substrings, ending up with g[0][6] being True because "abacaba" is a palindrome, and similarly g[2][4] being True for the substring "aca".

3. Next, we iterate over all possible pairs of indices (i, j) to find valid partitions. Our first loop with variable i will run from 0 to 4 (since we need at least two characters after the first partition for the remaining two palindromic substrings to be non-empty), and our nested loop with variable j will run from i+1 to 5.

4. For each pair (i, j), we check the 2D array g to see if g[0][i], g[i+1][j], and g[j+1][6] are all True.

5. When i is 2 (end of the first substring) and j is 4 (end of the second substring), we find that:

   • g[0][2] which represents "aba" is True,
   • g[3][4] which represents "ca" is False, and
   • g[5][6] which represents "ba" is False.

6. So the substring partitions "aba|ca|ba" are not palindromes collectively. We continue searching.

7. We eventually find that when i is 0 and j is 3, we get:

   • g[0][0] which represents "a" is True,
   • g[1][3] which represents "bac" is False, and
   • g[4][6] which represents "aba" is True.

   No palindromic partition here, so we keep searching.

8. Continuing this, we finally hit a combination when i is 2 and j is 5, resulting in the partitions "aba|cac|aba":

   • g[0][2] is True for "aba",
   • g[3][5] is True for "cac", and
   • g[6][6] is True for "a".

9. Thus, "abacaba" can be split into three palindromic substrings: "aba", "cac", and "a". At this point, we return True.

No further searching is needed; we have successfully found a split of the example string "abacaba" into three palindromic substrings using the outlined solution approach.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the provided code is O(n^3). Here's a breakdown:

• Building the g matrix involves a nested loop that compares each character to every other character, resulting in a O(n^2) time complexity.
• The nested loops where i ranges from 0 to n-3 and j ranges from i+1 to n-2 have a combined time complexity of O(n^2).
• Within the innermost loop, the code checks if three substrings (g[0][i], g[i+1][j], and g[j+1][-1]) are palindromes, which is an O(1) operation since the g matrix already contains this information.

These two processes are sequential, so we consider the larger of the two which is O(n^3) as the time complexity. This comes from the fact that for each potential partition point j, we consider each i and for each i and j, we check a constant-time condition.

Space Complexity

The space complexity of the code is O(n^2). This stems from the space required to store a 2D list (g) that maintains whether each substring from i to j is a palindrome. Since g is an n by n matrix, the space required is proportional to the square of the length of string s.

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