Subsets With Duplicates
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: Subsets With Duplicates * Group: 07. Recursion */ / * ============================================================================================ * PROBLEM: Subset (Generate Power Set) * ============================================================================================ * * --------------------- * 1. Problem Description * --------------------- * Given a set of distinct integers A, return all possible subsets (the Power Set). * * Key Requirements: * 1. Elements within each subset must be in non-descending order. * 2. The solution set must not contain duplicate subsets. * 3. The list of subsets itself must be sorted in lexicographical order. * (e.g., [1] comes before [1, 2], which comes before [2]). * * --------------------- * 2. Logical Breakdown & Mathematical Formulation * --------------------- * Let S be a set of N distinct elements. * The Power Set P(S) contains 2^N subsets. * * Strategy to satisfy ordering constraints: * 1. Internal Sort: Since elements in a subset must be non-descending, we MUST sort the * input array A at the very beginning. * * 2. Lexicographical Generation (Backtracking): * To ensure the subsets themselves are sorted ([], [1], [1, 2]...), we should use a * Depth-First Search (DFS) approach that: * - Adds the current subset to the result immediately. * - Iterates through the remaining valid elements to extend the current subset. * * Recurrence Logic: * function generate(index, currentSubset): * - Add copy of currentSubset to result list. * - Loop i from index to |A|-1: * - Add A[i] to currentSubset. * - generate(i + 1, currentSubset). * - Backtrack: Remove A[i] from currentSubset. * * This specific "Add-Loop-Recurse" pattern naturally produces the subsets in the * required lexicographical order if A is sorted. * * --------------------- * 3. Constraints & Complexity Analysis * --------------------- * Constraints: * 1 <= |A| <= 16 * * Time Complexity: * O(N * 2^N) * - There are 2^N subsets. * - Copying a subset takes O(N) on average. * - For N=16, 2^16 = 65,536. Total operations ~ 10^6, which is efficient. * * Space Complexity: * O(N) auxiliary stack space for recursion (excluding result storage). * * --------------------- * 4. Input / Output Format * --------------------- * Input: * - An integer array A. * * Output: * - A list of lists (2D vector) containing all subsets. * * --------------------- * 5. Examples * --------------------- * Input 1: A = [1] * Output 1: * [ * [], * [1] * ] * * Input 2: A = [1, 2, 3] * Output 2: * [ * [], * [1], * [1, 2], * [1, 2, 3], * [1, 3], * [2], * [2, 3], * [3] * ] * Note: [1, 3] appears after [1, 2, 3] because, lexicographically, the subset [1, 2...] * comes before [1, 3]. * * ============================================================================================ */
public class SubsetsWithDuplicates { public int solve(int A) { // TODO: Implement solution return 0; } }