B Closest Points To Origin
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: B Closest Points to Origin
* ============================================================================================
*
* ---------------------
* 1. Problem Description
* ---------------------
* Given a set of points (x, y) in A and an integer B.
* Find the B points that are closest to the origin (0, 0).
* Distance is calculated using the Euclidean formula: sqrt(x^2 + y^2).
*
* ---------------------
* 2. Logical Breakdown & Mathematical Formulation
* ---------------------
* To find the "top-k smallest" distances, we use a Max-Heap of size B.
*
* Step 1: Distance Metric
* For the purpose of comparison, we can use the squared Euclidean distance:
* $\(d^2 = x^2 + y^2\)$
* We avoid sqrt to maintain precision and improve performance.
*
* Step 2: Max-Heap Strategy
* 1. Initialize a Max-Heap to store point indices or objects, ordered by their \(d^2\).
* 2. Iterate through each point in A:
* a. Calculate its squared distance.
* b. Add the point to the Max-Heap.
* c. If the size of the heap exceeds B, remove the point with the maximum distance.
* 3. After processing all points, the heap contains the B closest points.
*
* ---------------------
* 3. Constraints & Complexity Analysis
* ---------------------
* Time Complexity:
* O(N log B)
* - We iterate through N points once.
* - Each heap operation (insertion/deletion) takes O(log B).
*
* Space Complexity:
* O(B) - To store the B closest points in the Priority Queue.
*
* ============================================================================================
/
public class BClosestPointsToOrigin {
public int[][] solve(int[][] a1, int A) {
// TODO: Implement solution
return new int[0][0];
}
}