Special Subsequences AG
Logical Breakdown
- [ ] Subproblem Identification:
- [ ] Optimal Substructure:
- [ ] Constraint Handling: (e.g., Modulo \(10^9 + 7\))
- [ ] Optimization: (Matrix Exponentiation if 'A' is huge)
Mathematical Rigor
State Definition
Let \(dp[i]\) be the state for the \(i\)-th subproblem.
Recurrence Relation
Visualization
Complexity Analysis
| Approach | Time Complexity | Space Complexity |
|---|---|---|
| Iterative DP | \(O(N)\) | \(O(1)\) |
Code Reference
package com.dsa.recursion;
import java.util.*;
/
* Problem: Special Subsequences AG
* Group: 07. Recursion
*/
/
* ============================================================================================
* PROBLEM: Special Subsequences "AG"
* ============================================================================================
*
* ---------------------
* 1. Problem Description
* ---------------------
* Given a string A consisting of uppercase English letters, count the total number of
* times the subsequence "AG" appears in the string.
*
* A subsequence is formed by deleting zero or more characters from the original string.
* Specifically, we are looking for indices (i, j) such that:
* 1. i < j
* 2. A[i] == 'A'
* 3. A[j] == 'G'
*
* NOTE: Return the answer modulo 10^9 + 7.
*
* ---------------------
* 2. Logical Breakdown & Optimization (Carry Forward)
* ---------------------
* Naive Approach:
* Iterate through every character. If it is 'A', iterate through the rest of the string
* to count 'G's.
* Time Complexity: O(N^2). For N=10^5, this is too slow (TLE).
*
* Optimized Approach (Carry Forward):
* We can solve this in a single pass O(N) by maintaining a running count of 'A's.
*
* Algorithm:
* 1. Initialize countA = 0 and ans = 0.
* 2. Iterate through the string from left to right:
* - If current char is 'A': Increment countA.
* - If current char is 'G': It means this 'G' can form a pair with all previous countA 'A's.
* Add countA to ans.
* - Apply modulo (ans % 10^9 + 7) at each addition to prevent overflow.
*
* Alternative (Suffix Sum):
* Iterate from right to left, counting 'G's. When an 'A' is found, add the count of 'G's to ans.
* Both approaches are mathematically equivalent.
*
* ---------------------
* 3. Constraints & Complexity Analysis
* ---------------------
* Constraints:
* 1 <= length(A) <= 10^5
*
* Time Complexity:
* O(N) - Single pass through the string.
*
* Space Complexity:
* O(1) - Using only integer variables for counters.
*
* ---------------------
* 4. Input / Output Format
* ---------------------
* Input:
* - A single string A.
*
* Output:
* - Return an integer denoting the count of "AG" subsequences modulo 10^9 + 7.
*
* ---------------------
* 5. Examples
* ---------------------
* Input 1: A = "ABCGAG"
* Traversal:
* - 'A': countA = 1
* - 'B': ignore
* - 'C': ignore
* - 'G': ans += 1 (Total 1)
* - 'A': countA = 2
* - 'G': ans += 2 (Total 3)
* Output 1: 3
*
* Input 2: A = "GAB"
* Traversal:
* - 'G': countA = 0 -> ans += 0
* - 'A': countA = 1
* - 'B': ignore
* Output 2: 0
*
* ============================================================================================
*/
public class SpecialSubsequencesAG {
public int solve(String A) {
// TODO: Implement solution
return 0;
}
}