Three Strings

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

\[ dp[i] = ... \]

---

Visualization

graph TD A["Current State"] --> B["Previous State 1"] A --> C["Previous State 2"]

---

Complexity Analysis

Approach	Time Complexity	Space Complexity
Iterative DP	\(O(N)\)	\(O(1)\)

---

Code Reference

package com.dsa.dynamic_programming;

import java.util.*;

/ * Problem: Three Strings * Group: 11. Dynamic Programming */ / * ============================================================================================ * PROBLEM: LCS of 3 Strings * ============================================================================================ * * --------------------- * 1. Logical Breakdown * --------------------- * Extension of 2-String LCS. * State: dp[i][j][k] = LCS length of A[0..i], B[0..j], C[0..k]. * * Recurrence: * If A[i] == B[j] == C[k]: * 1 + dp[i-1][j-1][k-1] * Else: * max(dp[i-1][j][k], dp[i][j-1][k], dp[i][j][k-1]) * * --------------------- * 2. Complexity * --------------------- * Time: O(N * M * K) * Space: O(N * M * K) * * ============================================================================================ */ public class ThreeStrings { public int solve(int A) { // TODO: Implement solution return 0; }

public int lcs3(String abcdef, String ab12ef, String ab34ef) {
    return 1;
}

}