1 CS Lunch Mary Allen Wilkes Wednesday 12:15 Kendade 307 2 Divide and Conquer Divide-and-conquer. Break up problem into several parts. Solve each part recursively. Combine solutions to sub-problems into overall solution. Most common usage. Break up problem of size n into two equal parts of size n/2. Solve two parts recursively. Combine two solutions into overall solution in linear time. 3 Closest Pair of Points Closest pair. Given n points in the plane, find a pair with smallest Euclidean distance between them. Fundamental geometric primitive. Graphics, computer vision, geographic information systems, molecular modeling, air traffic control. Brute force. Check all pairs of points p and q with Θ (n 2 ) comparisons. Slides14 - ClosestPairPoints.key - March 20, 2019
4 Closest Pair of Points 1-dimensional version 5-1 Closest Pair of Points 1-D version. Sort points For each point, find the distance between a point and the point that follows it. Remember the smallest. 5-2 Closest Pair of Points 1-D version. Cost Sort points For each point, find the distance between a point and the point that follows it. Remember the smallest. Slides14 - ClosestPairPoints.key - March 20, 2019
5-3 Closest Pair of Points 1-D version. Cost Sort points O(n log n) For each point, find the distance between a point and the point that follows it. Remember the smallest. 5-4 Closest Pair of Points 1-D version. Cost Sort points O(n log n) For each point, find the distance O(n) between a point and the point that follows it. Remember the smallest. 5-5 Closest Pair of Points 1-D version. Cost Sort points O(n log n) For each point, find the distance O(n) between a point and the point that follows it. Remember the smallest. Total is O(n log n) Slides14 - ClosestPairPoints.key - March 20, 2019
6-1 Closest Pair of Points Divide: draw vertical line L so that n/2 points on each side. L 6-2 Closest Pair of Points Divide: draw vertical line L so that n/2 points on each side. L 7-1 Closest Pair of Points Solve: find closest pair in each side recursively. L Slides14 - ClosestPairPoints.key - March 20, 2019
7-2 Closest Pair of Points Solve: find closest pair in each side recursively. L 12 7-3 Closest Pair of Points Solve: find closest pair in each side recursively. L 21 12 Closest Pair of Points 8 Combine: find closest pair with one point in each side. Return best of 3 solutions. How do we do this without comparing each point on left with each point on right??? L 8 21 12 Slides14 - ClosestPairPoints.key - March 20, 2019
9 Closest Pair of Points Let δ be the minimum between pair on left and pair on right If there exists a pair with one point in each side and whose distance < δ , find that pair. L 21 δ = min(12, 21) 12 10 Closest Pair of Points Observation: only need to consider points within δ of line L. δ L 21 δ = min(12, 21) 12 11 Closest Pair of Points Sort points in 2 δ -strip by their y coordinate. δ L 7 6 5 21 4 δ = min(12, 21) 3 12 2 1 Slides14 - ClosestPairPoints.key - March 20, 2019
12 Closest Pair of Points Only need to check distances of those within 15 positions in sorted list!!!! δ L 7 6 5 21 4 δ = min(12, 21) 3 12 2 1 13 Closest Pair of Points Let s i be the point in the 2 δ -strip, with j 39 the i th smallest y-coordinate. Claim. If |i – j| ≥ 16, then the distance 31 between s i and s j is at least δ . ½ δ 3 rows Proof: 30 ½ δ 29 No two points lie in same ½ δ -by- ½ δ ½ δ 28 i 27 box. Two points separated by at least 3 26 rows have distance ≥ 3( ½ δ ). 25 δ δ 14 Closest Pair of Points Let s i be the point in the 2 δ -strip, with the i th smallest y-coordinate. j 39 Claim. If |i – j| ≥ 16, then the distance between s i and s j is at least δ . 31 Proof: ½ δ No two points lie in same ½ δ -by- ½ δ 2 rows 30 ½ δ 29 box. ½ δ Two points separated by at least 3 i 28 27 rows have distance ≥ 3( ½ δ ). 26 If a point is within δ of point 27, it 25 must be in one of the blue boxes. There are only 15 blue boxes! δ δ Slides14 - ClosestPairPoints.key - March 20, 2019
15-1 Closest Pair Algorithm Closest-Pair(p 1 , …, p n ) { Compute separation line L such that half the points are on one side and half on the other side. δ 1 = Closest-Pair(left half) δ 2 = Closest-Pair(right half) δ = min( δ 1 , δ 2 ) Delete all points further than δ from separation line L Sort remaining points by y-coordinate. Scan points in y-order and calculate distance to all points with a y value within δ . If any of these distances is less than δ , update δ . return δ . } 15-2 Closest Pair Algorithm Cost Closest-Pair(p 1 , …, p n ) { Compute separation line L such that half the points are on one side and half on the other side. δ 1 = Closest-Pair(left half) δ 2 = Closest-Pair(right half) δ = min( δ 1 , δ 2 ) Delete all points further than δ from separation line L Sort remaining points by y-coordinate. Scan points in y-order and calculate distance to all points with a y value within δ . If any of these distances is less than δ , update δ . return δ . } 15-3 Closest Pair Algorithm Cost Closest-Pair(p 1 , …, p n ) { Compute separation line L such that half the points O(n log n) are on one side and half on the other side. δ 1 = Closest-Pair(left half) δ 2 = Closest-Pair(right half) δ = min( δ 1 , δ 2 ) Delete all points further than δ from separation line L Sort remaining points by y-coordinate. Scan points in y-order and calculate distance to all points with a y value within δ . If any of these distances is less than δ , update δ . return δ . } Slides14 - ClosestPairPoints.key - March 20, 2019
15-4 Closest Pair Algorithm Cost Closest-Pair(p 1 , …, p n ) { Compute separation line L such that half the points O(n log n) are on one side and half on the other side. 2T(n / 2) δ 1 = Closest-Pair(left half) δ 2 = Closest-Pair(right half) δ = min( δ 1 , δ 2 ) Delete all points further than δ from separation line L Sort remaining points by y-coordinate. Scan points in y-order and calculate distance to all points with a y value within δ . If any of these distances is less than δ , update δ . return δ . } 15-5 Closest Pair Algorithm Cost Closest-Pair(p 1 , …, p n ) { Compute separation line L such that half the points O(n log n) are on one side and half on the other side. 2T(n / 2) δ 1 = Closest-Pair(left half) δ 2 = Closest-Pair(right half) δ = min( δ 1 , δ 2 ) O(n) Delete all points further than δ from separation line L Sort remaining points by y-coordinate. Scan points in y-order and calculate distance to all points with a y value within δ . If any of these distances is less than δ , update δ . return δ . } 15-6 Closest Pair Algorithm Cost Closest-Pair(p 1 , …, p n ) { Compute separation line L such that half the points O(n log n) are on one side and half on the other side. δ 1 = Closest-Pair(left half) 2T(n / 2) δ 2 = Closest-Pair(right half) δ = min( δ 1 , δ 2 ) O(n) Delete all points further than δ from separation line L O(n log n) Sort remaining points by y-coordinate. Scan points in y-order and calculate distance to all points with a y value within δ . If any of these distances is less than δ , update δ . return δ . } Slides14 - ClosestPairPoints.key - March 20, 2019
15-7 Closest Pair Algorithm Cost Closest-Pair(p 1 , …, p n ) { Compute separation line L such that half the points O(n log n) are on one side and half on the other side. 2T(n / 2) δ 1 = Closest-Pair(left half) δ 2 = Closest-Pair(right half) δ = min( δ 1 , δ 2 ) O(n) Delete all points further than δ from separation line L O(n log n) Sort remaining points by y-coordinate. Scan points in y-order and calculate distance to all points with a y O(n) value within δ . If any of these distances is less than δ , update δ . return δ . } 15-8 Closest Pair Algorithm Cost Closest-Pair(p 1 , …, p n ) { Compute separation line L such that half the points O(n log n) are on one side and half on the other side. 2T(n / 2) δ 1 = Closest-Pair(left half) δ 2 = Closest-Pair(right half) δ = min( δ 1 , δ 2 ) O(n) Delete all points further than δ from separation line L O(n log n) Sort remaining points by y-coordinate. Scan points in y-order and calculate distance to all points with a y O(n) value within δ . If any of these distances is less than δ , update δ . T(n) ≤ 2T(n/2) + O(n log n) return δ . T(n) = O(n log 2 n) } Slides14 - ClosestPairPoints.key - March 20, 2019
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