Minimum Rectangles to Cover Points

Problem Description

You are given a 2D integer array points, where points[i] = [xi, yi]. You are also given an integer w. Your task is to cover all the given points with rectangles. Each rectangle has its lower end at some point (x1, 0) and its upper end at some point (x2, y2), where x1 <= x2, y2 >= 0, and the condition x2 - x1 <= w must be satisfied for each rectangle. A point is considered covered by a rectangle if it lies within or on the boundary of the rectangle. Return an integer denoting the minimum number of rectangles needed so that each point is covered by at least one rectangle.

Key Insights

• Each rectangle can cover multiple points as long as they fall within the width constraint (w).
• The problem can be approached by sorting the points based on their x-coordinates.
• A greedy strategy can be utilized to minimize the number of rectangles needed by extending the coverage of each rectangle as much as possible within the width limit.

Space and Time Complexity

Time Complexity: O(n log n) - due to sorting the points by x-coordinates. Space Complexity: O(1) - additional space usage is constant aside from the input storage.

Solution

To solve the problem, we can follow these steps:

1. Sort the points based on their x-coordinates.
2. Initialize a counter for rectangles needed.
3. Iterate through the sorted points and determine the range of x-coordinates that can be covered by the current rectangle considering the width w.
4. For each rectangle, find the maximum y-coordinate of the points within the current range.
5. Increment the rectangle counter each time a new rectangle is needed when the current point exceeds the range of the last placed rectangle.

This approach utilizes a greedy algorithm and ensures that we are covering the maximum number of points with each rectangle.

Code Solutions