#1745

Hard

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Palindrome Partitioning IV

The Palindrome Partitioning IV problem asks you to determine if a string can be split into three palindromic substrings.

string dynamic programming

Problem Statement

You are given a string s consisting of lowercase English letters. Your task is to determine whether it is possible to split this string into three non-empty palindromic substrings.

A string is a palindrome if it reads the same forward and backward. For instance, abcbdd can be split into a, bcb, and dd, each of which are palindromes. You need to return true if such a split is possible; otherwise, return false.

Examples

Example 1

Input: s = "abcbdd"

Output: true

"abcbdd" = "a" + "bcb" + "dd", and all three substrings are palindromes.

Example 2

Input: s = "bcbddxy"

Output: false

s cannot be split into 3 palindromes.

Constraints

3 <= s.length <= 2000
s consists only of lowercase English letters.

Solution Approach

Preprocessing Palindromes

Start by preprocessing the string to identify all possible palindromic substrings. This can be done using dynamic programming in O(n^2) time. Store the results in a 2D table where dp[i][j] indicates whether the substring s[i..j] is a palindrome.

State Transition DP

Use dynamic programming to break the problem into smaller subproblems. Let dp[i][j] represent whether the substring s[i..j] can be split into three palindromes. Transition between states by checking the validity of substrings and their palindromic properties.

Efficient Checking

To make the solution efficient, preprocess the string in a way that allows checking if a substring is a palindrome in O(1) time. This can be achieved using the 2D table generated earlier, minimizing redundant checks.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity depends on the approach used. Preprocessing palindromes takes O(n^2) time. The dynamic programming solution can be optimized to O(n^2) time for state transitions, given the preprocessed palindrome table. Space complexity is O(n^2) for storing the palindrome checks.

What Interviewers Usually Probe

Can the candidate efficiently preprocess palindrome checks in O(1) time for each substring?
Does the candidate understand the importance of dynamic programming for breaking down the problem into smaller subproblems?
Can the candidate discuss trade-offs in time and space complexity for this problem?

Common Pitfalls or Variants

Common pitfalls

Forgetting to preprocess palindrome checks, leading to inefficient solutions with redundant checks.
Overcomplicating the dynamic programming approach by not leveraging the preprocessed palindrome table.
Not correctly handling edge cases, such as strings that cannot be split into three palindromes.

Follow-up variants

Generalize the problem by allowing a different number of palindromic substrings (e.g., 2 or 4 substrings).
Extend the problem to handle a larger input size or strings with uppercase characters.
Optimize the solution further by reducing the space complexity of the palindrome check table.

FAQ

What is the optimal approach for solving Palindrome Partitioning IV?

The optimal approach involves preprocessing palindrome checks in O(n^2) time and then using dynamic programming to solve the problem efficiently in O(n^2) time and space.

How can I preprocess palindromes for efficient substring checking?

You can preprocess palindromes using a 2D dynamic programming table where dp[i][j] is true if the substring s[i..j] is a palindrome. This allows O(1) checks for each substring during state transitions.

What should I consider when optimizing the Palindrome Partitioning IV solution?

Consider optimizing space complexity by reducing the size of the palindrome table and avoiding redundant checks during state transitions.

How can I modify Palindrome Partitioning IV to handle a larger input size?

You can optimize by reducing space complexity and implementing more efficient dynamic programming transitions, while keeping the preprocessing step O(n^2).

What is the pattern behind the Palindrome Partitioning IV problem?

The problem follows the state transition dynamic programming pattern, where subproblems are solved based on earlier solutions and palindromic substrings are preprocessed for efficient checks.

terminal

Solution

Solution 1: Dynamic Programming

We define $f[i][j]$ to indicate whether the substring of $s$ from the $i$-th character to the $j$-th character is a palindrome, initially $f[i][j] = \textit{true}$.

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class Solution:
    def checkPartitioning(self, s: str) -> bool:
        n = len(s)
        f = [[True] * n for _ in range(n)]
        for i in range(n - 1, -1, -1):
            for j in range(i + 1, n):
                f[i][j] = s[i] == s[j] and (i + 1 == j or f[i + 1][j - 1])
        for i in range(n - 2):
            for j in range(i + 1, n - 1):
                if f[0][i] and f[i + 1][j] and f[j + 1][-1]:
                    return True
        return False

Continue Practicing

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