Magician And Chocolates
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: Magician and Chocolates (Greedy with Max-Heap)
* ============================================================================================
*
* ---------------------
* 1. Problem Description
* ---------------------
* Given N bags with Bi chocolates. In each of the A units of time:
* - The kid eats all chocolates in bag i.
* - The magician refills bag i with floor(Bi / 2) chocolates.
* Task: Maximize total chocolates eaten. Return result modulo 10^9 + 7.
*
* ---------------------
* 2. Logical Breakdown & Mathematical Formulation
* ---------------------
* This is a Greedy problem. To maximize the sum, we must always pick the bag that
* currently contains the maximum number of chocolates.
*
* Algorithm:
* 1. Initialization:
* - Use a Max-Priority Queue (PriorityQueue with Collections.reverseOrder()).
* - Insert all N bag capacities into the heap.
*
* 2. Simulation (For A iterations):
* - Extract the maximum element maxChoc from the heap.
* - Add maxChoc to the running total (using modulo 10^9 + 7).
* - Calculate the refill amount: refill = floor(maxChoc / 2).
* - Insert refill back into the heap.
*
* 3. Finalization:
* - Return the total accumulated chocolates.
*
* ---------------------
* 3. Constraints & Complexity Analysis
* ---------------------
* Constraints:
* 1 <= N <= 10^5
* 1 <= A <= 10^5
* 1 <= Bi <= 10^9
*
* Time Complexity:
* O(N log N + A log N)
* - Building the heap: O(N log N).
* - A operations of extract-max and insert: O(A log N).
*
* Space Complexity:
* O(N) to store the heap elements.
* ============================================================================================
/
public class MagicianAndChocolates {
public int solve(int A) {
// TODO: Implement solution
return 0;
}
public int nchoc(int i, int[] b1) {
return 1;
}
}