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Different Palindromic Subsequences

Certification: Advanced Level Accuracy: 0% Submissions: 0 Points: 15

Write a C program to count the number of different non-empty palindromic subsequences in a given string. Since the answer may be very large, return it modulo 10^9 + 7. 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.

Example 1
    
  
  • Input: s = "bccb"
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  • Output: 6
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  • Explanation: 
    • The 6 different palindromic subsequences are 'b', 'c', 'c', 'b', 'cc', 'bccb'. 
    • Note that 'bcb' is not a valid subsequence since the characters must maintain their original order. 
    • Therefore, the total count is 6.
    •
Example 2
    
  
  • Input: s = "abcdabcdabcdabcdabcdabcdabcdabcddcbadcbadcbadcbadcbadcbadcbadcba"
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  • Output: 104860361
    • 
  
  • Explanation: 
    • This is a complex string with many possible palindromic subsequences. 
    • We need to count all different palindromic subsequences efficiently. 
    • The result is 104860361 after taking modulo 10^9 + 7.
    •
Constraints
    
  
  • 1 <= s.length <= 1000
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  • s[i] is either 'a', 'b', 'c', or 'd'
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  • The answer should be returned modulo 10^9 + 7
    • 
  
  • Time Complexity: O(n^3)
    • 
  
  • Space Complexity: O(n^2)
    •
Dynamic Programming CapgeminiLTIMindtree
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Hint
Solution

Solution Hints

    
  
  • Use dynamic programming with 3D table dp[i][j][c] representing count of palindromes from index i to j starting and ending with character c
    • 
  
  • For each character 'a', 'b', 'c', 'd', calculate palindromes separately
    • 
  
  • Base case: single characters form palindromes of length 1
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  • For longer substrings, if s[i] == s[j] == c, then dp[i][j][c] = 2 + sum of all dp[i+1][j-1][x] for all characters x
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  • If characters don't match at ends, copy from smaller ranges
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  • Use modular arithmetic to avoid overflow
    •

Steps to solve by this approach:

 Step 1: Create a 3D DP table dp[i][j][c] where i and j are indices and c represents characters 'a', 'b', 'c', 'd'
 Step 2: Initialize base cases - single characters form palindromes of count 1
 Step 3: For each substring length from 2 to n, fill the DP table
 Step 4: For each character c, if s[i] == s[j] == c, then palindromes = 2 + sum of all inner palindromes
 Step 5: If only one end matches character c, copy count from the substring excluding the non-matching end
 Step 6: If neither end matches c, copy count from the inner substring
 Step 7: Sum up all dp[0][n-1][c] for all characters c and return the result modulo 10^9 + 7

Submitted Code :