#1637

Easy

LeetCode Problem Workspace

Widest Vertical Area Between Two Points Containing No Points

Find the maximum width of a vertical area between points with no points inside using array sorting efficiently.

Problem Statement

You are given a list of n points on a 2D plane, where each point is represented as points[i] = [xi, yi]. Your task is to determine the maximum width of a vertical area between any two points such that no other points lie inside this area. A vertical area extends infinitely along the y-axis, and points on the boundaries are not considered inside.

Return the width of the widest vertical area between two points. For example, given points = [[8,7],[9,9],[7,4],[9,7]], the widest vertical area has a width of 1. The solution requires careful handling of the array plus sorting pattern to efficiently identify gaps between consecutive points along the x-axis.

Examples

Example 1

Input: points = [[8,7],[9,9],[7,4],[9,7]]

Output: 1

Both the red and the blue area are optimal.

Example 2

Input: points = [[3,1],[9,0],[1,0],[1,4],[5,3],[8,8]]

Output: 3

Example details omitted.

Constraints

n == points.length
2 <= n <= 105
points[i].length == 2
0 <= xi, yi <= 109

Solution Approach

Sort Points by X-coordinate

Begin by extracting the x-coordinates from all points and sort them in ascending order. Sorting aligns points along the x-axis, making it straightforward to evaluate consecutive gaps and ensuring we correctly apply the array plus sorting pattern.

Compute Consecutive Differences

Iterate through the sorted x-coordinates and calculate the differences between each pair of consecutive points. Track the maximum difference, which represents the widest vertical area with no points inside. This step leverages the insight that the largest gap occurs between sorted neighbors.

Return Maximum Width

After computing all consecutive differences, return the maximum value. This ensures the final output corresponds to the widest vertical area between two points, maintaining O(N log N) time and O(log N) space complexity.

Complexity Analysis

Metric	Value
Time	O(N \log N)
Space	O(\log N)

Sorting the x-coordinates requires O(N log N) time and O(log N) space due to the sorting algorithm stack. Iterating through consecutive differences is O(N), so the overall time complexity is O(N log N) and space complexity remains O(log N).

What Interviewers Usually Probe

Sorting the points is the first step; neglecting it can lead to incorrect width calculation.
Focusing on consecutive x-coordinates reveals the maximum gap; random comparisons fail to leverage the pattern.
Boundary points do not count inside the vertical area; including them will produce off-by-one errors.

Common Pitfalls or Variants

Common pitfalls

Attempting nested loops over all points increases time complexity unnecessarily.
Confusing y-coordinates with x-coordinates can lead to incorrect area widths.
Forgetting to exclude boundary points from the vertical area can yield wrong results.

Follow-up variants

Find the widest horizontal area with no points inside by sorting y-coordinates instead.
Determine the k largest vertical gaps rather than just the maximum width.
Handle points with duplicate x-coordinates and ensure the area width remains zero for overlapping points.

FAQ

What is the main pattern used in Widest Vertical Area Between Two Points Containing No Points?

The problem follows the array plus sorting pattern, where sorting x-coordinates allows efficient identification of maximum gaps between points.

Can duplicate x-coordinates affect the solution?

Yes, duplicates result in zero-width gaps, so the algorithm must handle consecutive identical x-values correctly.

Is it necessary to consider y-coordinates?

No, only x-coordinates determine vertical area width; y-values do not influence the gap calculation.

What is the optimal time complexity for this problem?

Sorting x-coordinates gives O(N log N) time, and calculating differences is O(N), so the overall complexity is O(N log N).

How does GhostInterview help avoid common mistakes here?

GhostInterview emphasizes sorting first and focusing on consecutive differences, preventing off-by-one errors and unnecessary nested iterations.

terminal

Solution

Solution 1

#### Python3

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class Solution:
    def maxWidthOfVerticalArea(self, points: List[List[int]]) -> int:
        points.sort()
        return max(b[0] - a[0] for a, b in pairwise(points))

Solution 2

#### Python3

1

2

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class Solution:
    def maxWidthOfVerticalArea(self, points: List[List[int]]) -> int:
        points.sort()
        return max(b[0] - a[0] for a, b in pairwise(points))

Continue Practicing

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