Given an array of points on the X-Y plane, return the k closest points to the origin (0, 0).
Input: first line = points as x:y comma-separated, second = k. Output each point on a separate line.
Example
Input
1:3,-2:2
1
Output
-2 2
Reasoning
The input points are parsed as (1, 3) and (-2, 2) from the string "1:3,-2:2".
The distance of each point to the origin is calculated using the Euclidean distance formula: d=x2+y2. For the given points, the distances are 12+32=10 and (−2)2+22=8.
The point with the smallest distance to the origin is selected, which is (-2, 2) since 8<10.
The selected point is output as "-2 2", which matches the given output format.
The problem "K Closest Points to Origin" involves geometry and mathematical concepts, specifically the distance formula. The distance between two points (x1,y1) and (x2,y2) in a 2D plane is given by (x2−x1)2+(y2−y1)2. In this case, we want to find the distance from each point to the origin (0, 0), which simplifies to x2+y2.
To solve this problem, it's essential to understand how to calculate distances and compare them. The concept of sorting is also crucial, as we need to rank the points based on their distances to the origin. Additionally, we should be familiar with data structures such as arrays or lists, which will be used to store the points and their corresponding distances.
The problem also involves algorithmic thinking, specifically the ability to iterate over a set of points, calculate distances, and select the closest points. Understanding time and space complexity is also important, as we need to ensure our solution is efficient in terms of both execution time and memory usage.
Algorithm/Approach
The general approach to solve this type of problem involves using a sorting-based algorithm. We can calculate the distance of each point to the origin, store the points and their distances in a data structure, and then sort the points based on their distances. The k closest points can be selected from the sorted list.
Step-by-Step Strategy
To implement the solution, follow these steps:
Parse the input string into a list of points, where each point is represented as a pair of coordinates (x, y).
Calculate the distance of each point to the origin using the distance formula.
Store the points and their distances in a data structure, such as a list of tuples or a dictionary.
Sort the points based on their distances to the origin.
Select the k closest points from the sorted list.
Output each of the k closest points on a separate line.
Common Pitfalls
When implementing the solution, watch out for the following:
Incorrectly calculating the distance to the origin.
Failing to handle edge cases, such as points with identical distances.
Using an inefficient sorting algorithm or data structure.
Not considering the time and space complexity of the solution.
Time & Space Complexity
The expected time complexity of the solution is O(n log n) due to the sorting operation, where n is the number of points. The space complexity is O(n) for storing the points and their distances. However, the actual complexity may vary depending on the specific implementation and the choice of data structures and algorithms.