QuestionQuestion

Given n points on the x-axis line (in some arbitrary order), find the closest pair of points in O(nlogn) time.

1)Show that the problem can be solved using sorting.
2)Give a divide and conquer algorithm that does not use sorting. The advantage of this approach is that it generalizes to points in higher dimensions.

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Part 1 – Sorting
Using sorting, the problem can be solved in two steps:
• we apply a sorting algorithm to order the points by their x-coordinate
• we linearly iterate through the sorted list, compute the distance between adjacent points and update the minimum with every occasion
For this algorithm, the running time complexity is also O(nlogn)+O(n)= O(nlogn) (assuming we use a sorting algorithm which runs in O(nlogn) time like MergeSort or HeapSort)....

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