Quickhull Algorithm for Convex Hull

Given a set of points, a Convex hull is the smallest convex polygon containing all the given points.

👁 Convex-Hull

Input : points[] = {{0, 3}, {1, 1}, {2, 2}, {4, 4},
 {0, 0}, {1, 2}, {3, 1}, {3, 3}};
Output : The points in convex hull are:
 (0, 0) (0, 3) (3, 1) (4, 4)
Input : points[] = {{0, 3}, {1, 1}
Output : Not Possible
There must be at least three points to form a hull.
Input : points[] = {(0, 0), (0, 4), (-4, 0), (5, 0), 
 (0, -6), (1, 0)};
Output : (-4, 0), (5, 0), (0, -6), (0, 4)

We have discussed following algorithms for Convex Hull problem. Convex Hull | Set 1 (Jarvis’s Algorithm or Wrapping)Convex Hull | Set 2 (Graham Scan) The QuickHull algorithm is a Divide and Conquer algorithm similar to QuickSort. Let a[0...n-1] be the input array of points. Following are the steps for finding the convex hull of these points.

1. Find the point with minimum x-coordinate lets say, min_x and similarly the point with maximum x-coordinate, max_x.
2. Make a line joining these two points, say L. This line will divide the whole set into two parts. Take both the parts one by one and proceed further.
3. For a part, find the point P with maximum distance from the line L. P forms a triangle with the points min_x, max_x. It is clear that the points residing inside this triangle can never be the part of convex hull.
4. The above step divides the problem into two sub-problems (solved recursively). Now the line joining the points P and min_x and the line joining the points P and max_x are new lines and the points residing outside the triangle is the set of points. Repeat point no. 3 till there no point left with the line. Add the end points of this point to the convex hull.

Below is C++ implementation of above idea. The implementation uses set to store points so that points can be printed in sorted order. A point is represented as a pair. 

The points in Convex Hull are:
(0, 0) (0, 3) (3, 1) (4, 4) 

Time Complexity: The analysis is similar to Quick Sort. On average, we get time complexity as O(n Log n), but in worst case, it can become O(n²)

space complexity : O(n) 

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