Question

Consider the algorithm to find the closest pair of points in the plane. Let's say you wanted to generalize the algorithm to find the two closest pairs of points in the plane given a set of (unsorted) points (p1, py. Give an algorithm for finding the two distances for this pair. In the step to conquer the two subproblems, you must explain why your algorithm is guaranteed to find the correct result. You do not need to specify the best possible algorithm, but your overall algorithm must run in O(n log n) time

Answer 1

We are given an array of n points in the plane, and the problem is to find out the closest pair of points in the array.

Algorithm to find closest pair of points(O(nlogn))
step 1) First we sort all points according to x coordinates.
step 2) Divide set all points in two set of half size.
step 3) Find the smallest distances in both subarrays recursively.
step 4) find the minimum of two smallest distances say it d.
step 5) Create an array points[] that stores all points which are at most d distance away from the middle line dividing the two sets.
step 6) Find the smallest distance in points[].
step 7) Return the minimum of d and the smallest distance calculated in above step 6.