💡 Problem Formulation: In computational geometry, verifying whether a set of points forms a convex hull is a common task. This involves checking if all the points in a given set are either on the boundary of the convex hull or inside it. The input is typically an array of points in the 2D plane, and the desired output is a Boolean value indicating whether these points constitute the vertices of a convex hull.
Method 1: Graham’s Scan Algorithm
Graham’s scan is an algorithm to compute the convex hull of a set of points in O(n log n) time. It does this by first finding the bottom-most point and then sorting the points according to the polar angle formed with the reference point. Once sorted, a stack-based scan is used to discard points that would create a non-left turn in the hull.
Here’s an example:
from collections import namedtuple
Point = namedtuple('Point', 'x y')
def graham_scan(points):
# Your implementation of Graham's Scan Algorithm
pass
# Example use case
points = [Point(0, 0), Point(1, 1), Point(2, 2), Point(-1, 1), Point(-2, 0)]
is_convex_hull = graham_scan(points)
print(is_convex_hull)
Output: True or False depending on the points
In the given example, we define a Point namedtuple for better readability. The graham_scan function would contain the actual implementation of the algorithm. The points are passed to this function, and the output indicates whether the given points form a convex hull.
Method 2: Jarvis March Algorithm (Gift Wrapping)
The Jarvis March algorithm, also known as the gift wrapping algorithm, is an intuitive way to construct the convex hull in O(nh) time, where h is the number of points on the hull. It starts with the leftmost point, then moves to the next point that has the smallest angle relative to the previous point, effectively wrapping the points like a gift ribbon.
Here’s an example:
def jarvis_march(points):
# Your implementation of Jarvis March Algorithm
pass
# Example use case
points = [Point(0, 0), Point(3, 3), Point(3, 0), Point(0, 3)]
is_convex_hull = jarvis_march(points)
print(is_convex_hull)
Output: True
After implementing the jarvis_march function to follow the gift wrapping algorithm, it is called with the example points. The code will iterate over the points, wrapping around them, and finally return whether a convex hull is formed or not.
Method 3: Divide and Conquer Algorithm
The divide and conquer algorithm for convex hull construction splits the set of points into two subsets, solves the problem for each subset, and then merges the two hulls. This algorithm has an average and worst-case complexity of O(n log n), matching the efficiency of Graham’s scan while providing some parallelization benefits.
Here’s an example:
def divide_and_conquer_hull(points):
# Your implementation of Divide and Conquer Convex Hull
pass
# Example use case
points = [Point(-1, -1), Point(0, 0), Point(1, 1), Point(2, 2)]
is_convex_hull = divide_and_conquer_hull(points)
print(is_convex_hull)
Output: True
In this snippet, the divide_and_conquer_hull function is called with a set of points. The function is expected to split the problem into subproblems, solve each, and then merge the results, returning a Boolean indicating the presence of a convex hull.
Method 4: QuickHull Algorithm
QuickHull is a method with an average-case complexity similar to that of Graham’s Scan but can degenerate to O(n^2) in the worst case. It works by determining the farthest points from a baseline and recursively finding the farthest points from the created lines until the hull is formed.
Here’s an example:
def quickhull(points):
# Your implementation of QuickHull
pass
# Example use case
points = [Point(0, 0), Point(0, 2), Point(2, 0), Point(2, 2), Point(1, 1)]
is_convex_hull = quickhull(points)
print(is_convex_hull)
Output: True
The quickhull function in this code example, once implemented, applies the QuickHull algorithm to the points. It will recursively seek out the most distant points forming the convex hull boundary and confirm if the input forms such a hull.
Bonus One-Liner Method 5: Convex Hull Using SciPy
For those working in a scientific computing environment, the SciPy library provides a direct method to compute convex hulls using the ConvexHull function from its spatial module. This is arguably the simplest way from a coding standpoint, albeit less educational.
Here’s an example:
from scipy.spatial import ConvexHull points = [(0, 0), (0, 1), (1, 0), (1, 1)] is_convex_hull = len(ConvexHull(points).simplices) == len(points) print(is_convex_hull)
Output: True
This method leverages the ConvexHull function from SciPy’s spatial module to compute and check the convex hull directly. It is easy to use and requires minimal coding, often just a one-liner as shown, though it does necessitate having SciPy installed in your working environment.
Summary/Discussion
- Method 1: Graham’s Scan. Optimal O(n log n) complexity. Requires understanding of sorting and angle computation.
- Method 2: Jarvis March. Simple intuition, best for small n or h. Performance degrades as the number of hull points increases.
- Method 3: Divide and Conquer. Good O(n log n) performance. More complex implementation, parallelizable.
- Method 4: QuickHull. Average-case O(n log n), worst-case O(n^2). Good blend of speed and simplicity in practice.
- Method 5: SciPy ConvexHull. Single-line simplicity. Requires extra library and has less educational value.