Palindrome Substrings II in C++ (Time Complexity: O(n^2))

Given a string, count the number of palindromic contiguous substrings in the string.

The substrings with different start indexes or end indexes are counted as different substrings even they consist of same characters.

Example:

Input:  "abbcbc"

Output: 9

Explanation: ["a", "b", "b", "c", "b", "c", "bb", "bcb", "cbc"]

Understanding the Problem

The core challenge of this problem is to identify all contiguous substrings of a given string that are palindromes. A palindrome is a string that reads the same forward and backward. The significance of this problem lies in its applications in text processing, DNA sequence analysis, and other fields where pattern recognition is crucial.

Potential pitfalls include missing substrings that are palindromes or counting the same substring multiple times. Misconceptions might arise from not understanding that substrings with different start or end indexes are considered different even if they consist of the same characters.

Approach

To solve this problem, we can use a dynamic programming approach. The naive solution would involve checking all possible substrings, which would be inefficient. Instead, we can use a 2D table to store whether a substring is a palindrome and build up our solution from smaller substrings to larger ones.

Naive Solution

The naive solution involves generating all possible substrings and checking each one for being a palindrome. This approach has a time complexity of O(n^3) and is not optimal for large strings.

Optimized Solution

We can optimize the solution using dynamic programming. The idea is to use a 2D table where dp[i][j] is true if the substring from index i to j is a palindrome. We initialize the table for substrings of length 1 and 2, and then use these results to build up the solution for longer substrings.

Algorithm

1. Initialize a 2D table dp where dp[i][j] is true if the substring s[i:j+1] is a palindrome.

2. All substrings of length 1 are palindromes.

3. Check substrings of length 2 and mark them as palindromes if both characters are the same.

4. For substrings longer than 2, use the results of smaller substrings to determine if they are palindromes.

5. Count all true values in the dp table to get the number of palindromic substrings.

Code Implementation

#include <iostream>
#include <vector>
#include <string>

using namespace std;

// Function to count palindromic substrings
int countPalindromicSubstrings(const string& s) {
    int n = s.length();
    vector<vector<bool>> dp(n, vector<bool>(n, false));
    int count = 0;

    // Every single character is a palindrome
    for (int i = 0; i < n; ++i) {
        dp[i][i] = true;
        count++;
    }

    // Check for palindromes of length 2
    for (int i = 0; i < n - 1; ++i) {
        if (s[i] == s[i + 1]) {
            dp[i][i + 1] = true;
            count++;
        }
    }

    // Check for palindromes of length greater than 2
    for (int length = 3; length <= n; ++length) {
        for (int i = 0; i < n - length + 1; ++i) {
            int j = i + length - 1;
            if (s[i] == s[j] && dp[i + 1][j - 1]) {
                dp[i][j] = true;
                count++;
            }
        }
    }

    return count;
}

int main() {
    string s = "abbcbc";
    cout << "Number of palindromic substrings: " << countPalindromicSubstrings(s) << endl;
    return 0;
}

Complexity Analysis

The time complexity of the optimized solution is O(n^2) because we are filling an n x n table. The space complexity is also O(n^2) due to the storage of the dp table. This is a significant improvement over the naive O(n^3) solution.

Edge Cases

Potential edge cases include:

Empty string: Should return 0.
String with all identical characters: Every substring is a palindrome.
String with no palindromic substrings longer than 1.

These cases can be tested to ensure the robustness of the solution.

Testing

To test the solution comprehensively, consider the following test cases:

Simple cases: "a", "aa", "ab"
Complex cases: "abbcbc", "racecar", "aabb"
Edge cases: "", "aaaa", "abcde"

Using a testing framework like Google Test can help automate and manage these tests effectively.

Thinking and Problem-Solving Tips

When approaching such problems, it's essential to:

Break down the problem into smaller, manageable parts.
Consider both naive and optimized solutions.
Use dynamic programming to build solutions from smaller subproblems.
Practice similar problems to improve problem-solving skills.

Conclusion

Understanding and solving problems like counting palindromic substrings is crucial for developing strong algorithmic skills. By breaking down the problem, considering different approaches, and optimizing the solution, we can tackle complex challenges effectively. Practice and continuous learning are key to mastering such problems.

Additional Resources

For further reading and practice, consider the following resources: