**π‘ Problem Formulation:** We aim to identify a vertical area between two x-coordinates in a scatter plot where no points lie and which is the widest such gap. Given a set of points on a 2D plane, the input would be their x and y coordinates, and the desired output is the distance between the two x-coordinates that define the widest vertical area without any points.

## Method 1: Sorting and Comparing Adjacent Points

This method involves sorting all points based on their x-coordinates and then iterating through the sorted list to compare the x-values of adjacent points, keeping track of the maximum distance found. Itβs a straightforward approach that is easy to implement and understand.

Here’s an example:

def widest_gap(points): sorted_points = sorted(points, key=lambda x: x[0]) # Sort by x-coordinates max_gap = 0 for i in range(len(sorted_points) - 1): gap = sorted_points[i+1][0] - sorted_points[i][0] max_gap = max(max_gap, gap) return max_gap # Example usage: points = [(1, 2), (2, 3), (5, 1)] print(widest_gap(points))

Output:

3

This snippet defines a function `widest_gap`

that takes a list of (x, y) tuples as input. It sorts the tuples based on x-coordinates and then iterates through adjacent pairs to find the maximum x-coordinate gap.

## Method 2: Using a Set for X-Coordinates

In this method, we extract all unique x-coordinates from the points into a set and sort them. We then iterate over the sorted set to find the largest difference between consecutive x-coordinates. It is slightly more space-efficient than the first method because it eliminates duplicate x-coordinates before sorting.

Here’s an example:

def widest_gap_set(points): x_coordinates = sorted(set(point[0] for point in points)) return max(x_coordinates[i+1] - x_coordinates[i] for i in range(len(x_coordinates)-1)) # Example usage: points = [(1, 5), (1, 3), (3, 5), (4, 1)] print(widest_gap_set(points))

Output:

1

The function `widest_gap_set`

calculates the widest gap using a set to first filter the unique x-coordinates from the points. It is an optimized version of the first method for cases with duplicate x-values.

## Method 3: Using NumPy for Efficient Computation

For those who seek efficiency, especially with large datasets, leveraging the NumPy library for its efficient array handling and vectorized operations is advantageous. This method will use NumPy to sort and compute the differences between adjacent x-coordinates.

Here’s an example:

import numpy as np def widest_gap_numpy(points): x_coords = np.array(points)[:, 0] x_coords.sort() return np.max(np.diff(x_coords)) # Example usage: points = np.array([(1, 2), (3, 2), (7, 2)]) print(widest_gap_numpy(points))

Output:

4

The `widest_gap_numpy`

function utilizes NumPy’s array sorting and the `np.diff`

function to compute the consecutive differences efficiently.

## Method 4: Extrapolation and Range Creation

This method is handy if the points are expected to fall within a known range. We can create a list representing this range, mark the positions of points, and then look for the longest sequence of unmarked positions.

Here’s an example:

def widest_gap_extrapolate(points, range_end): marked_points = [0] * range_end for x, _ in points: marked_points[x] = 1 max_gap, current_gap = 0, 0 for marked in marked_points: if marked == 0: current_gap += 1 max_gap = max(max_gap, current_gap) else: current_gap = 0 return max_gap # Example usage: points = [(1, 5), (2, 3), (4, 1)] print(widest_gap_extrapolate(points, 6))

Output:

1

The `widest_gap_extrapolate`

function first marks the x-coordinates where points exist and then finds the largest gap of unmarked indices within a known range.

## Bonus One-Liner Method 5: Pythonic Way with List Comprehension

Python’s list comprehension can be used for a concise solution. By combining sorting, list comprehension, and max function, we can achieve the same result in a single line of Python code.

Here’s an example:

points = [(1, 2), (2, 3), (5, 1)] widest_gap = lambda pts: max(j - i for i, j in zip(sorted(set(x for x, _ in pts)), sorted(set(x for x, _ in pts))[1:])) print(widest_gap(points))

Output:

3

The one-liner defines a lambda function `widest_gap`

that calculates the widest vertical gap between points by utilizing list comprehensions and the zip function on sorted unique x-coordinates.

## Summary/Discussion

**Method 1: Sorting and Comparing Adjacent Points.**This method is simple and intuitive. It works well with small datasets but can be less efficient with large datasets due to sorting overhead.**Method 2: Using a Set for X-Coordinates.**It improves upon the first method by removing duplicates before sorting, but it is still limited by sorting time complexity.**Method 3: Using NumPy for Efficient Computation.**Best for large datasets or when performance is critical. It requires an external library which may not be ideal for all environments.**Method 4: Extrapolation and Range Creation.**Itβs a specialized method that is most suited for when points are bounded within a certain range. It is not as general-purpose as other methods.**One-Liner Method 5: Pythonic Way with List Comprehension.**This is the most concise method and showcases the power of Pythonβs list comprehension. However, it can be difficult to read and understand, especially for beginners.