Computational Geometry Questions Long
The algorithm for finding the closest pair of points in a set using Computational Geometry techniques is known as the Closest Pair algorithm. This algorithm is based on the divide and conquer strategy and has a time complexity of O(n log n), where n is the number of points in the set.
Here is a step-by-step description of the algorithm:
1. Sort the points in the set based on their x-coordinate. This can be done using any efficient sorting algorithm such as merge sort or quicksort. Let's assume the sorted points are stored in an array P.
2. Divide the set of points into two equal halves based on the x-coordinate. Let's call the left half L and the right half R.
3. Recursively find the closest pair of points in each half. This can be done by applying the Closest Pair algorithm on L and R separately.
4. Let d be the minimum distance between the closest pair of points found in the left half (L) and the right half (R).
5. Create a strip of width 2d in the middle of the set of points. This strip is defined as the set of points whose x-coordinate lies within d distance from the middle line. Sort the points in the strip based on their y-coordinate.
6. Iterate through each point in the strip and compare its distance to the next 7 points in the strip. This is because the maximum number of points that can be within d distance from a given point is 7.
7. If any pair of points in the strip has a distance less than d, update d to be the distance between them.
8. Return the minimum distance d as the closest pair of points in the set.
The Closest Pair algorithm takes advantage of the fact that the closest pair of points can only exist in the left half, right half, or the strip. By recursively dividing the set of points and merging the results, the algorithm efficiently finds the closest pair of points.