AnotherSequenceProblem
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: AnotherSequenceProblem * Group: 07. Recursion */ / * ============================================================================================ * PROBLEM: Another Sequence Problem * ============================================================================================ * * --------------------- * 1. Problem Description * --------------------- * Given a sequence defined by the recurrence relation: * f(A) = f(A-1) + f(A-2) + f(A-3) + A * * Calculate the Ath term of the sequence. * * Given Base Cases: * f(0) = 1 * f(1) = 1 * f(2) = 2 * * --------------------- * 2. Logical Breakdown & Mathematical Formulation * --------------------- * Let T(n) be the nth term of this sequence. * * Recurrence Relation: * T(n) = T(n-1) + T(n-2) + T(n-3) + n for n >= 3 * * Base Cases (Stopping Conditions): * 1. If n == 0 : Return 1 * 2. If n == 1 : Return 1 * 3. If n == 2 : Return 2 * * --------------------- * 3. Constraints & Complexity Analysis * --------------------- * Constraints: * 0 <= A <= 20 * * Time Complexity: * - Naive Recursion: The growth rate is similar to the Tribonacci sequence (approx O(1.84^A)). * For A = 20, 1.84^20 is approximately 180,000 operations. * This is well within the standard limit of 10^8 operations per second. * - Memoization (Optional): Would reduce this to O(A). * * Space Complexity: * O(A) - Recursion stack depth. * * Data Type Note: * Since A <= 20, the result fits comfortably within a standard 32-bit signed integer. * * --------------------- * 4. Input / Output Format * --------------------- * Input: * - A single integer A. * * Output: * - Return an integer denoting the Ath term. * * --------------------- * 5. Examples * --------------------- * Input 1: A = 3 * Calculation: * f(3) = f(2) + f(1) + f(0) + 3 * = 2 + 1 + 1 + 3 * = 7 * Output 1: 7 * * Input 2: A = 2 * Calculation: Base case given. * Output 2: 2 * * ============================================================================================ */ public class AnotherSequenceProblem { public int solve(int A) { // TODO: Implement solution return 0; } }