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Different Palindromic Subsequences

Certification: Advanced Level Accuracy: 100% Submissions: 1 Points: 15

Write a Java program to count the number of different palindromic subsequences in a given string. A subsequence is a sequence that can be derived from another sequence by deleting some or no elements without changing the order of the remaining elements. A palindrome is a string that reads the same backward as forward. The answer should be computed modulo 10^9 + 7 because the result could be very large.

Note: The empty string is not considered a palindromic subsequence.

Example 1
    
  
  • Input: s = "bccb"
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  • Output: 6
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  • Explanation: The different palindromic subsequences are: "b", "c", "bb", "cc", "bcb", "bccb". "b" appears twice in the string, but is counted only once. "c" appears twice in the string, but is counted only once.
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Example 2
    
  
  • Input: s = "abcdabcba"
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  • Output: 15
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  • Explanation: The different palindromic subsequences are: "a", "b", "c", "d", "aa", "bb", "cc", "aba", "aca", "ada", "aba", "aca", "bcb", "abcba", "abcdabcba". "a", "b", and "c" each appear multiple times but are counted only once.
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Constraints
    
  
  • The input string length will not exceed 1000
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  • The input string consists of lowercase English letters only
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  • Time Complexity: O(n^2), where n is the length of the string
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  • Space Complexity: O(n^2)
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  • The result should be computed modulo 10^9 + 7
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ArraysDynamic Programming HCL TechnologiesAdobe
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Hint
Solution

Solution Hints

    
  
  • Use dynamic programming to solve this problem
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  • Define dp[i][j] as the number of different palindromic subsequences in the substring s[i...j]
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  • Base case: dp[i][i] = 1 for each character
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  • If s[i] == s[j], the number of palindromic subsequences will depend on whether s[i] appears within the substring s[i+1...j-1]
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  • If s[i] != s[j], we need to consider palindromic subsequences in s[i...j-1] and s[i+1...j], avoiding double counting
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  • Consider edge cases carefully when characters appear multiple times
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  • Remember to take modulo 10^9 + 7 for each calculation
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Steps to solve by this approach:

 Step 1: Create a 2D DP array where dp[i][j] represents the number of different palindromic subsequences in the substring s[i...j].

 Step 2: Initialize the base case: dp[i][i] = 1 for each character (each character is a palindrome).

 Step 3: Fill the DP table by considering substrings of increasing length.

 Step 4: For each substring s[i...j], if s[i] != s[j], calculate dp[i][j] = dp[i][j-1] + dp[i+1][j] - dp[i+1][j-1] to avoid double counting.

 Step 5: If s[i] == s[j], the result depends on whether s[i] appears within the substring s[i+1...j-1].

 Step 6: Check three cases: no occurrence, one occurrence, or multiple occurrences of s[i] in the middle, and calculate accordingly.

 Step 7: Take modulo 10^9 + 7 at each step to prevent integer overflow.

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