#131

Medium

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Palindrome Partitioning

Find all possible palindrome partitioning of a string using backtracking and dynamic programming.

Problem Statement

Given a string s, your task is to partition it into substrings such that each substring is a palindrome. Return all possible ways to partition the string satisfying this condition. For example, with s = 'aab', the correct partitions are [['a','a','b'], ['aa','b']].

You can solve this problem using a dynamic programming approach combined with backtracking to explore all possible partitions. Each substring must be checked for being a palindrome, and the problem can be framed using state transitions where the valid partitions are constructed progressively.

Examples

Example 1

Input: s = "aab"

Output: [["a","a","b"],["aa","b"]]

Example details omitted.

Example 2

Input: s = "a"

Output: [["a"]]

Example details omitted.

Constraints

1 <= s.length <= 16
s contains only lowercase English letters.

Solution Approach

State Transition Dynamic Programming

Start by creating a dynamic programming table where dp[i] holds all the valid palindrome partitions of the substring s[0:i]. Initialize the table with dp[0] as an empty list. From there, use the state transitions to expand the table, checking whether the current substring is a palindrome before adding it to the partition.

Backtracking to Generate All Partitions

Once the dynamic programming table is built, use backtracking to explore all possible partitions. For each valid palindrome ending at index i, backtrack to generate partitions, ensuring that every substring within the partition is a palindrome. This results in all possible ways to partition the string into palindromic substrings.

Optimization with Memoization

To further optimize, apply memoization during the palindrome check step. This avoids redundant checks and speeds up the solution, especially as the size of the string grows. Using a helper function to store already checked substrings will improve efficiency, reducing unnecessary recomputations.

Complexity Analysis

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

The time complexity is dominated by the dynamic programming state transitions and backtracking steps, making it O(2^n), where n is the length of the string. The space complexity depends on the space needed for storing the dynamic programming table and the call stack during backtracking, which is O(n^2).

What Interviewers Usually Probe

Do you understand how to use dynamic programming with backtracking in palindrome partitioning?
Can you efficiently identify palindromes and generate all partitions using state transitions?
Will you be able to optimize the solution with memoization to handle larger inputs?

Common Pitfalls or Variants

Common pitfalls

Forgetting to check for palindrome substrings before adding them to partitions can result in incorrect outputs.
Not using dynamic programming effectively to store intermediate results can lead to redundant computations, slowing down the solution.
Failing to optimize the algorithm with memoization can cause performance issues for larger strings, especially with repeated checks.

Follow-up variants

Palindrome Partitioning II (min cuts): Find the minimum number of cuts needed for a palindrome partition.
Partition Strings into Palindromes (non-overlapping): Partition the string with no overlap and each substring being a palindrome.
Palindrome Partitioning III: Partition the string with the condition of maximizing the number of palindromic partitions.

FAQ

What is the primary approach to solve Palindrome Partitioning?

Palindrome Partitioning is typically solved by combining dynamic programming to store valid partitions and backtracking to explore all possible ways to split the string into palindromes.

How can you optimize Palindrome Partitioning for larger inputs?

Using memoization for the palindrome check step and dynamic programming for storing intermediate results can significantly optimize the solution, making it more efficient for larger strings.

How does backtracking help in Palindrome Partitioning?

Backtracking helps by generating all possible partitions of the string. It ensures that each partition consists only of palindromic substrings, allowing all valid partitioning options to be explored.

What is the role of dynamic programming in Palindrome Partitioning?

Dynamic programming plays a crucial role by storing intermediate results, specifically whether substrings are palindromes, which makes generating partitions more efficient and avoids redundant computations.

What are some common mistakes when solving Palindrome Partitioning?

Common mistakes include not checking substrings for palindromes before adding them to partitions, failing to use dynamic programming to avoid redundant computations, and not optimizing the algorithm with memoization.

terminal

Solution

Solution 1: Preprocessing + DFS (Backtracking)

We can use dynamic programming to preprocess whether any substring in the string is a palindrome, i.e., $f[i][j]$ indicates whether the substring $s[i..j]$ is a palindrome.

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class Solution:
    def partition(self, s: str) -> List[List[str]]:
        def dfs(i: int):
            if i == n:
                ans.append(t[:])
                return
            for j in range(i, n):
                if f[i][j]:
                    t.append(s[i : j + 1])
                    dfs(j + 1)
                    t.pop()

        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 f[i + 1][j - 1]
        ans = []
        t = []
        dfs(0)
        return ans

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