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Leetcode 1637. Widest Vertical Area Between Two Points Containing No Points

Problem Statement

Given n points on a 2D plane, where points[i] = [xi, yi], return the widest vertical area between two points such that no points are inside the area. A vertical area is defined as an area of width w where no points lie inside the area. Formally, a vertical area is defined by two parallel lines x = a and x = b where there are no points with x-coordinate in the range (a, b), and a < b.

Clarifying Questions

Before diving into the solution, let’s clarify the problem requirements:

  1. Are there any constraints on the values of xi and yi?
    • The coordinates are integers, and there might be constraints like -10^9 <= xi, yi <= 10^9.
  2. Can there be duplicate points?
    • No, each point [xi, yi] is unique as implied by the problem statement.
  3. What should be returned if only one point is given?
    • Since there’s no area to measure with a single point, the answer should be 0.

Strategy

To find the widest vertical area, we can leverage the following strategy:

  1. Extract and Sort x-coordinates: Sort the points based on their x-coordinates.
  2. Compute Maximum Gap: Calculate the differences between successive x-coordinates to find the maximum vertical area width.

Code

import java.util.Arrays;

public class Solution {
    public int maxWidthOfVerticalArea(int[][] points) {
        // Edge case: when the number of points is less than 2
        if (points.length < 2) {
            return 0;
        }

        // Extracting the x-coordinates
        int[] xCoords = new int[points.length];
        for (int i = 0; i < points.length; i++) {
            xCoords[i] = points[i][0];
        }

        // Sorting the x-coordinates
        Arrays.sort(xCoords);

        // Finding the maximum gap between consecutive x-coordinates
        int maxGap = 0;
        for (int i = 1; i < xCoords.length; i++) {
            int gap = xCoords[i] - xCoords[i - 1];
            if (gap > maxGap) {
                maxGap = gap;
            }
        }

        return maxGap;
    }

    public static void main(String[] args) {
        Solution solution = new Solution();
        int[][] points = { {8,7}, {9,9}, {7,4}, {9,7} };
        System.out.println(solution.maxWidthOfVerticalArea(points)); // Output: 1
    }
}

Time Complexity

  1. Sorting: The major time-consuming step is sorting the x-coordinates, which takes O(n log n) time complexity where n is the number of points.
  2. Finding the Maximum Gap: This step is O(n), as we iterate through the sorted coordinates.

Therefore, the overall time complexity is dominated by the sorting step and is O(n log n).

Summary

This solution effectively handles the problem by focusing on sorting the x-coordinates and then finding the maximum gap between successive coordinates. This approach ensures that we cover all potential vertical areas efficiently.

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