Product Of 3
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: Product of 3 (Running Top-K Product)
* ============================================================================================
*
* ---------------------
* 1. Problem Description
* ---------------------
* Given an array A. For every index i (from 0 to N-1), find the product of the
* three largest integers found in the range A[0...i].
* - If the current range has fewer than 3 elements, return -1 for that index.
*
* ---------------------
* 2. Logical Breakdown & Mathematical Formulation
* ---------------------
* We need to maintain the "Top 3" largest elements dynamically as we iterate
* through the array.
*
* Algorithm (Min-Heap approach):
* 1. Initialization:
* - Create an empty Min-Priority Queue (Min-Heap).
* - Create a result array B of size N.
*
* 2. Iterative Processing:
* - For each element A[i]:
* a. Add A[i] to the Min-Heap.
* b. If the heap size exceeds 3, remove the smallest element (poll).
* - This ensures the heap always contains the 3 largest elements seen so far.
* c. Check heap size:
* - If size < 3: Set B[i] = -1.
* - If size == 3:
* - Extract all 3 elements (or peek/iterate) to calculate their product.
* - Set B[i] = val1 * val2 * val3.
* - (Note: In Java, you can iterate over the PriorityQueue or poll 3 times
* and re-add them).
*
* ---------------------
* 3. Constraints & Complexity Analysis
* ---------------------
* Constraints:
* 1 <= N <= 10^5
* 1 <= A[i] <= 10^3
*
* Time Complexity:
* O(N * log(3)) ≈ O(N)
* - For each of the N elements, we perform heap operations. Since the heap
* size is capped at 3, log(3) is a constant.
*
* Space Complexity:
* O(N) for the output array and O(1) (specifically O(3)) for the heap.
* ============================================================================================
/
public class ProductOf3 {
public int[] solve(int[] A) {
// TODO: Implement solution
return new int[0];
}
}