08 - Divide and conquer algorithms in computational geometry- finding the closest pair of points

08 - Divide and conquer algorithms in computational geometry- finding the closest pair of points

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2/8/08 - Divide and conquer algorithms. .. Lecture: Divide and conquer algorithms in computational geometry: fnding the closest pair oF points Reading: Chapter 5.4 cs482-02-08-08-Audio.mp4 Closest pair oF points Given n numbers x 1 , x 2 , . .. , x n ±ind i, j such that |x i – x j | is minimized. O(nlogn): sort them. y 1 < . .. y n Compute every y i+1 – y i , take the min. Given n points {(x i , y i ): i = 1 . .. n} ±ind i, j that minimize: ±irst idea: Step 1: Sort them by x coord. Sort them by y coord. 11-1 11
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Let a be the median x coordinate. Take the line L. x = a. Let Q = {pts left of L} R = {pts right of L} Step 2: Solve the Q, R subproblems Set = min{ δ Q , δ R } Step 1. Sorting (twice) O(nlogn) Step 2: Solving Q, R 2f(n/2)
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Unformatted text preview: f(m):= runtime on m points Step 3: Dealing with the strip. (nlogn) f(n) = 2f(n/2) + O(nlogn) nlog 2 (n). g(m):= runtime on m pre-sorted (by x and y) points g(m)=2g(m/2) + Step 3 f(m) = O(nlogn) + g(m) Lemma: If p 1 , p 2 , . .. p n are points in a vertical strip of width 2 , where = min dist between pairs on the same side of the strips midline, sorted by y coord, and if i, j such that p i p j &lt; , then the closest such pair satisFes |i-j| 16 Make a grid of side length /2 Key observation: 11-2 No grid cell contains more than one point. Diameter of a grid cell is /2* 2 11-3...
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08 - Divide and conquer algorithms in computational geometry- finding the closest pair of points

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