#3111

Medium

auto_awesomeGreedy choice plus invariant validation

LeetCode Problem Workspace

Minimum Rectangles to Cover Points

Find the minimum number of rectangles needed to cover all points, given constraints on width and position.

Problem Statement

You are given a 2D array of distinct points, where each point is represented by an x and y coordinate. You must use rectangles to cover these points, with the condition that the rectangle's width cannot exceed a given integer w. Each rectangle has its left side at x=0 and extends to some x2 such that x2 - x1 <= w. You are tasked with determining the minimum number of rectangles needed to cover all the points.

A point is covered by a rectangle if it lies within or on the boundary of the rectangle. Only the x-coordinates of the points matter, as the y-values do not affect the rectangle's placement. The problem emphasizes greedy strategies and invariant validation, focusing on optimizing the rectangle placements based on the x-axis alone.

Examples

Example 1

Input: points = [[2,1],[1,0],[1,4],[1,8],[3,5],[4,6]], w = 1

Output: 2

The image above shows one possible placement of rectangles to cover the points:

Example 2

Input: points = [[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6]], w = 2

Output: 3

The image above shows one possible placement of rectangles to cover the points:

Example 3

Input: points = [[2,3],[1,2]], w = 0

Output: 2

The image above shows one possible placement of rectangles to cover the points:

Constraints

1 <= points.length <= 105
points[i].length == 2
0 <= xi == points[i][0] <= 109
0 <= yi == points[i][1] <= 109
0 <= w <= 109
All pairs (xi, yi) are distinct.

Solution Approach

Sort Points by X-coordinate

First, sort the points based on their x-coordinate. This step is crucial because it enables an efficient way of checking which points can fit within a rectangle. By processing points in sorted order, you can easily group them into rectangles as you iterate.

Greedy Rectangle Placement

Place a rectangle starting at the first uncovered point. Extend the rectangle as far right as possible, ensuring that all points within the rectangle satisfy the width constraint. Continue this process until all points are covered.

Increment Rectangles Count

Each time you place a new rectangle, increment the count. The greedy approach ensures that the rectangles are placed in an optimal manner, covering as many points as possible with each rectangle.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity primarily depends on sorting the points, which takes O(n log n), where n is the number of points. After sorting, the greedy placement process is linear, O(n). Therefore, the overall time complexity is O(n log n). Space complexity is O(n) due to storing the points array.

What Interviewers Usually Probe

Check if the candidate understands the significance of sorting the points by the x-coordinate.
Observe if the candidate uses the greedy method efficiently to cover points.
Evaluate whether the candidate correctly handles the width constraint during rectangle placement.

Common Pitfalls or Variants

Common pitfalls

Overcomplicating the problem by trying to account for y-coordinates, which are not needed.
Not recognizing that sorting is the first step, which could lead to inefficient rectangle placements.
Failing to increment the rectangle count correctly after each placement, leading to inaccurate results.

Follow-up variants

Handling edge cases where points are already aligned on a single vertical line.
Adjusting the rectangle width dynamically instead of using a fixed w value.
Optimizing the solution for larger point sets, potentially using a divide-and-conquer approach.

FAQ

What is the minimum number of rectangles needed for the 'Minimum Rectangles to Cover Points' problem?

The minimum number of rectangles is the fewest number of rectangles needed to cover all points while respecting the width constraint.

What pattern should be used to solve the 'Minimum Rectangles to Cover Points' problem?

The solution uses a greedy approach combined with sorting the points based on their x-coordinate to efficiently place rectangles.

How does sorting the points help in solving the 'Minimum Rectangles to Cover Points' problem?

Sorting the points by their x-coordinate allows for a greedy strategy to place rectangles, covering the maximum number of points in each step.

Why do the y-values not matter in the 'Minimum Rectangles to Cover Points' problem?

The rectangles only need to span the x-axis within the given width constraint, so the y-values do not affect the solution.

What is the time complexity of the 'Minimum Rectangles to Cover Points' problem?

The time complexity is O(n log n) due to the sorting step, followed by a linear scan to place rectangles.

terminal

Solution

Solution 1: Greedy + Sorting

According to the problem description, we do not need to consider the height of the rectangles, only the width.

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class Solution:
    def minRectanglesToCoverPoints(self, points: List[List[int]], w: int) -> int:
        points.sort()
        ans, x1 = 0, -1
        for x, _ in points:
            if x > x1:
                ans += 1
                x1 = x + w
        return ans

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