N Max Pair Combinations
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.heaps;
import java.util.*;
/* * ============================================================================================ * PROBLEM: N Max Pair Combinations * ============================================================================================ * * --------------------- * 1. Logical Breakdown & Mathematical Formulation * --------------------- * Given arrays A and B of size N. We need N largest sums of (A[i] + B[j]). * * Step 1: Sorting * - Sort both A and B in descending order. * - The maximum possible sum is always A[0] + B[0]. * * Step 2: Max-Heap with Index Tracking * - Use a Max-Heap to store triples: {sum, index_A, index_B}. * - Use a Set (HashSet) to keep track of visited index pairs (i, j) to avoid * pushing the same combination multiple times. * * Step 3: Iterative Extraction * 1. Push {A[0] + B[0], 0, 0} into the Max-Heap. * 2. Repeat N times: * a. Extract the maximum sum combination {S, i, j} from the heap. * b. Add S to the result list. * c. Potential next candidates are {A[i+1] + B[j], i+1, j} and {A[i] + B[j+1], i, j+1}. * d. For each candidate, if the indices are within bounds (i, j < N) and * not visited, push them into the heap and mark as visited. * * --------------------- * 2. Complexity Analysis * --------------------- * Time Complexity: * O(N log N) * - Sorting: O(N log N). * - N extractions/insertions in heap: O(N log N). * - Set operations: O(N). * * Space Complexity: * O(N) to store the heap elements and the visited set. * * --------------------- * 3. Base Case Verification * --------------------- * - N=1: Only A[0]+B[0] is extracted. * - If all elements are equal: The set prevents redundant processing of the same index pairs. * ============================================================================================ / public class NMaxPairCombinations { public int solve(int A) { // TODO: Implement solution return 0; }
public int[] solve(int[] a3, int[] b3) {
return new int[0];
}
}