This problem requires selecting the k points nearest to the origin by calculating Euclidean distances and efficiently organizing results. Solutions often use a max-heap to track closest points or Quickselect for partial sorting. Understanding distance comparisons and array manipulations is key for optimal performance.
Problem Statement
You are given an array of points where each point is represented as points[i] = [xi, yi] on a 2D plane. Your task is to return the k points closest to the origin (0, 0) based on Euclidean distance.
The Euclidean distance between a point (x, y) and the origin is calculated as sqrt(xx + yy). You may return the points in any order, and the answer is guaranteed to be unique with respect to point selection, though order can vary.
Examples
Example 1
Input: points = [[1,3],[-2,2]], k = 1
Output: [[-2,2]]
The distance between (1, 3) and the origin is sqrt(10). The distance between (-2, 2) and the origin is sqrt(8). Since sqrt(8) < sqrt(10), (-2, 2) is closer to the origin. We only want the closest k = 1 points from the origin, so the answer is just [[-2,2]].
Example 2
Input: points = [[3,3],[5,-1],[-2,4]], k = 2
Output: [[3,3],[-2,4]]
The answer [[-2,4],[3,3]] would also be accepted.
Constraints
- 1 <= k <= points.length <= 104
- -104 <= xi, yi <= 104
Solution Approach
Max-Heap Approach
Use a max-heap of size k to store the closest points seen so far. Iterate through all points, and if a point is closer than the current farthest in the heap, replace it. This ensures we always maintain the k closest points efficiently.
Quickselect Partitioning
Apply the Quickselect algorithm to partially sort the points based on their squared Euclidean distance. Partition the array until the first k points are the closest, avoiding a full sort and reducing average time complexity.
Sorting Entire Array
Compute the squared distance for all points and sort the entire array based on these distances. Select the first k points after sorting. This is simpler to implement but may be slower for large inputs.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
Max-heap maintains O(n log k) time and O(k) space. Quickselect achieves average O(n) time and O(1) extra space. Full sort requires O(n log n) time and O(n) space. Trade-offs involve implementation simplicity versus performance on large datasets.
What Interviewers Usually Probe
- Does the candidate correctly compute Euclidean distances without using square roots unnecessarily?
- Do they choose an optimal approach between heap, Quickselect, and full sort for k closest points?
- Can they handle edge cases such as multiple points at the same distance or k equal to array length?
Common Pitfalls or Variants
Common pitfalls
- Using sqrt unnecessarily, increasing computation time instead of comparing squared distances.
- Not maintaining max-heap of size k, causing extra space usage or wrong points selection.
- Failing to handle cases where multiple points have identical distances to the origin.
Follow-up variants
- Find k farthest points from origin using a min-heap or modified Quickselect.
- Find k closest points to an arbitrary point (x0, y0) instead of the origin.
- Find k closest points in 3D space using Euclidean distance in three dimensions.
How GhostInterview Helps
- GhostInterview identifies the most efficient strategy between heap, Quickselect, and sorting for your input size.
- It provides step-by-step reasoning for comparing distances without unnecessary square root calculations.
- GhostInterview highlights common pitfalls like handling equal distances and maintaining correct heap size for accurate results.
Topic Pages
Related GhostInterview Pages
- LeetCode Interview Copilot - Use GhostInterview as a live solver when you want direct help with LeetCode-style coding questions.
- Coding Interview Assistant - See how GhostInterview supports array, string, linked list, graph, and tree interview workflows.
- How GhostInterview Works - Review the screenshot, reasoning, and answer flow before using the solver in a live interview.
FAQ
What is the best approach for K Closest Points to Origin?
Using a max-heap for dynamic tracking of k closest points or Quickselect for partial sorting usually provides optimal performance.
Should I compute sqrt for Euclidean distance?
No, comparing squared distances is sufficient and avoids extra computation, maintaining correct order for selection.
Can the answer be in any order?
Yes, the points can be returned in any order as long as the closest k points are selected.
What is the time complexity for this problem?
Max-heap approach runs in O(n log k), Quickselect averages O(n), and full sort requires O(n log n).
How do I handle multiple points at the same distance?
Any subset of points at the same distance can be chosen as long as k points are returned; order does not matter.
Need direct help with K Closest Points to Origin instead of spending more time grinding it?
Download GhostInterview when you want a LeetCode solver, not another long practice loop. Capture K Closest Points to Origin from a screenshot, get the answer path and complexity, and use supported stealth workflows that stay outside captured layers.
Capture the prompt fast instead of rewriting the problem by hand.
Get the solution path, trade-offs, and complexity summary in one pass.
Stay outside captured layers on supported screen-share workflows.
Stay in the same pattern family
Find the k most frequent elements from an array using efficient algorithms like hashing and sorting.
Open problem page#215 Kth Largest Element in an ArrayFind the kth largest element in an unsorted array using optimal approaches like Quickselect or heaps.
Open problem page#1738 Find Kth Largest XOR Coordinate ValueCompute the kth largest XOR coordinate in a 2D matrix using prefix sums, bit manipulation, and optimized selection techniques.
Open problem page