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Quickhull is an efficient algorithm used to find the convex hull of a set of points in a plane. It works by recursively dividing the point set into two subsets and finding the convex hull for each subset.
Quickhull Algorithm for Convex hull:- Base Case:
- If the set of points has 0 or 1 elements, it is already the convex hull.
- Find the Endpoints:
- Find the two points with the minimum and maximum x-coordinates. These points are guaranteed to be part of the convex hull.
- Divide the Set:
- Divide the point set into two subsets: points lying to the left of the line formed by the two endpoints, and points lying to the right.
- Find Hull Recursively:
- Recursively find the convex hull for the points lying on the left and right sides of the line.
- Merge Hulls:
- Merge the convex hulls of the left and right subsets to form the final convex hull.
Python Implementation for Quickhull Algorithm for Convex hull:
Python
def find_distance(p1, p2, p):
"""
Calculate the distance between a point p and the line formed by p1 and p2.
"""
return (p[1] - p1[1]) * (p2[0] - p1[0]) - (p2[1] - p1[1]) * (p[0] - p1[0])
def quickhull(points):
"""
Find the convex hull of a set of points using the Quickhull algorithm.
Parameters:
points (list): A list of (x, y) coordinate tuples representing the points.
Returns:
list: A list of (x, y) coordinate tuples representing the convex hull.
"""
if len(points) <= 1:
return points
def hull(p1, p2, points):
"""
Find the points in the input set that lie outside the triangle formed by p1, p2, and the convex hull.
"""
if not points:
return []
farthest_point = max(points, key=lambda p: abs(find_distance(p1, p2, p)))
next_points = [p for p in points if find_distance(p1, farthest_point, p) > 0]
return hull(p1, farthest_point, next_points) + [farthest_point] + hull(farthest_point, p2, next_points)
min_point = min(points)
max_point = max(points)
convex_hull = hull(min_point, max_point, points) + hull(max_point, min_point, points)
return sorted(set(convex_hull))
# Example usage:
points = [(0, 3), (1, 1), (2, 2), (4, 4), (0, 0), (1, 2), (3, 1), (3, 3)]
convex_hull = quickhull(points)
print("Convex Hull Points:", convex_hull)
OutputConvex Hull Points: [(0, 0), (0, 3), (2, 2), (3, 1), (3, 3)]
Complexity Analysis:- Time Complexity: The time complexity of Quickhull is typically O(n * log(n)), where n is the number of points in the input set. However, in the worst-case scenario, it can be O(n^2) if poorly chosen pivot points result in unbalanced partitions.
- Space Complexity: Quickhull has a space complexity of O(n) due to the recursive calls and the storage of intermediate points.
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