#2472

Hard

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Maximum Number of Non-overlapping Palindrome Substrings

Find the maximum number of non-overlapping palindromic substrings of at least length k in a string using dynamic programming.

two pointers string dynamic programming greedy

Problem Statement

You are given a string s and a positive integer k. Your task is to find the maximum number of non-overlapping palindromic substrings that have a length of at least k.

A palindrome is a string that reads the same forwards and backwards. The substrings you select must be palindromic, and they cannot overlap with each other. Return the maximum number of such substrings in an optimal selection.

Examples

Example 1

Input: s = "abaccdbbd", k = 3

Output: 2

We can select the substrings underlined in s = "abaccdbbd". Both "aba" and "dbbd" are palindromes and have a length of at least k = 3. It can be shown that we cannot find a selection with more than two valid substrings.

Example 2

Input: s = "adbcda", k = 2

Output: 0

There is no palindrome substring of length at least 2 in the string.

Constraints

1 <= k <= s.length <= 2000
s consists of lowercase English letters.

Solution Approach

Identify Palindromic Substrings

Start by identifying all possible palindromic substrings of length at least k. Use dynamic programming to check each substring and store results to avoid recomputation.

Use Two Pointers to Ensure Non-overlap

Once palindromic substrings are identified, employ a two-pointer technique to select non-overlapping substrings. As you select one, adjust the pointers to avoid overlap with previously selected substrings.

Optimize with Dynamic Programming

Optimize the solution using dynamic programming to track the maximum number of valid substrings, ensuring that the selections respect the condition of non-overlap and the length constraint.

Complexity Analysis

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

The time and space complexity depend on the dynamic programming approach used to identify palindromes and the efficiency of the two-pointer technique. Expect a time complexity of O(n^2) for palindrome identification, with additional complexity for dynamic programming and state transitions.

What Interviewers Usually Probe

Candidate efficiently applies dynamic programming to solve the problem.
Demonstrates understanding of state transitions and two-pointer technique.
Can identify and optimize the problem by carefully considering substring overlap.

Common Pitfalls or Variants

Common pitfalls

Failing to efficiently check for palindromes, leading to a slow solution.
Not ensuring non-overlapping substrings, causing incorrect selections.
Overcomplicating the dynamic programming approach without considering time complexity.

Follow-up variants

Modify the problem by requiring palindromic substrings to be of an exact length, not just at least k.
Limit the total number of palindromic substrings to a fixed number.
Extend the problem to handle uppercase letters or non-ASCII characters.

FAQ

How do I approach the Maximum Number of Non-overlapping Palindrome Substrings problem?

Start by identifying all possible palindromic substrings of length at least k, then use dynamic programming and a two-pointer technique to select non-overlapping substrings.

What is the time complexity for this problem?

The time complexity is typically O(n^2) due to palindrome checks and dynamic programming state transitions, though optimizations can be applied.

Can this problem be solved without dynamic programming?

Dynamic programming is highly effective for this problem, but alternative methods may exist, though they may be less efficient.

What should I consider to avoid overlapping palindromic substrings?

Use a two-pointer approach to track the positions of selected substrings and ensure they do not overlap.

How can I optimize the dynamic programming solution for this problem?

Focus on minimizing recomputation by storing previously computed results for palindromic substrings and maintaining state transitions for efficient selection.

terminal

Solution

Solution 1: Preprocessing + Memoization Search

First, preprocess the string $s$ to get $dp[i][j]$, which represents whether the substring $s[i,..j]$ is a palindrome.

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class Solution:
    def maxPalindromes(self, s: str, k: int) -> int:
        @cache
        def dfs(i):
            if i >= n:
                return 0
            ans = dfs(i + 1)
            for j in range(i + k - 1, n):
                if dp[i][j]:
                    ans = max(ans, 1 + dfs(j + 1))
            return ans

        n = len(s)
        dp = [[True] * n for _ in range(n)]
        for i in range(n - 1, -1, -1):
            for j in range(i + 1, n):
                dp[i][j] = s[i] == s[j] and dp[i + 1][j - 1]
        ans = dfs(0)
        dfs.cache_clear()
        return ans

Continue Practicing

#3302 find the lexicographically smallest valid sequence #2486 append characters to string to make subsequence #2522 partition string into substrings with values at most k