Maximum Array Sum After B Negations
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;
/*
* ============================================================================================
* PROBLEM: Maximum Array Sum After B Negations
* ============================================================================================
*
* ---------------------
* 1. Problem Description
* ---------------------
* Given an array A and an integer B. You must perform exactly B negations (A[i] = -A[i]).
* Goal: Maximize the total sum of the array.
*
* ---------------------
* 2. Logical Breakdown & Mathematical Formulation
* ---------------------
* To maximize the sum, we must follow a greedy strategy:
* * Step 1: Greedy Choice
* - In each step, we should negate the smallest element currently in the array.
* - If the smallest element is negative, negating it makes it positive, providing the
* largest possible increase to the sum.
* - If all elements are positive, negating the smallest positive number results in
* the smallest possible decrease to the sum.
*
* Step 2: Efficiency via Min-Heap
* - Since we need the minimum element repeatedly across B operations, a Min-Heap
* is optimal.
* * Algorithm:
* 1. Build a Min-Heap from all elements in array A.
* 2. Repeat B times:
* a. Extract the minimum element minVal.
* b. Negate it: newVal = -minVal.
* c. Insert newVal back into the Min-Heap.
* 3. After B operations, extract all elements from the heap and sum them up.
*
* ---------------------
* 3. Constraints & Complexity Analysis
* ---------------------
* Constraints:
* 1 <= N <= 5 * 10^5, 1 <= B <= 5 * 10^6
* * Time Complexity:
* O(N + B log N)
* - Heapification: O(N).
* - B operations of extract-min and insert: O(B log N).
*
* Space Complexity:
* O(N) to store the elements in the Priority Queue.
* ============================================================================================
/
public class MaximumArraySumAfterBNegations {
public int solve(int[] a4, int A) {
// TODO: Implement solution
return 0;
}
}