Lets Party

Logical Breakdown

• [ ] Subproblem Identification:
• [ ] Optimal Substructure:
• [ ] Constraint Handling: (e.g., Modulo \(10^9 + 7\))
• [ ] Optimization: (Matrix Exponentiation if 'A' is huge)

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Mathematical Rigor

State Definition

Let \(dp[i]\) be the state for the \(i\)-th subproblem.

Recurrence Relation

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

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Visualization

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

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Complexity Analysis

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

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Code Reference

package com.dsa.dynamic_programming;

import java.util.*;

/ * Problem: Lets Party * Group: 11. Dynamic Programming */ / * ============================================================================================ * PROBLEM: Let's Party * ============================================================================================ * * --------------------- * 1. Problem Description * --------------------- * There are A people at a party. Each person can either: * 1. Dance alone. * 2. Dance in a pair with one other person. * * Find the total number of distinct ways they can organize themselves. * * --------------------- * 2. Logical Breakdown & Mathematical Formulation * --------------------- * Consider the N-th person. They have two choices: * * Choice 1: Dance Alone. * - The remaining (N-1) people can arrange themselves in dp[N-1] ways. * * Choice 2: Pair Up. * - The N-th person can pick any of the other (N-1) people to partner with. * - Number of partners = (N-1). * - After picking a partner, (N-2) people remain. They can arrange in dp[N-2] ways. * - Contribution: (N-1) * dp[N-2]. * * Recurrence Relation: * dp[i] = dp[i-1] + (i-1) * dp[i-2] * * Base Cases: * dp[1] = 1 (1 person: {1}) * dp[2] = 2 (2 people: {1, 2} or {1-2}) * * --------------------- * 3. Constraints & Complexity * --------------------- * Time Complexity: O(A). * Space Complexity: O(A) or O(1). * * ============================================================================================ */ public class LetsParty { public int solve(int A) { // TODO: Implement solution return 0; }

public int solveParty(int i) {
    return 0;
}

}