LecXX4 - COT 6936 Topics in Algorithms Giri Narasimhan ECS...

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1 4/1/10 COT 6936 1 COT 6936: Topics in Algorithms Giri Narasimhan ECS 254A / EC 2443; Phone: x3748 [email protected] http://www.cs.fiu.edu/~giri/teach/COT6936_S10.html https://online.cis.fiu.edu/portal/course/view.php?id=427 COT 6936 2 Closest Pair Problem ± Input : Set of points S in the plane ± Output : The closest pair of points in S ± Naïve Solution : O(n 2 ) time ± Divide-&-Conquer : ± T(n) = 2T(n/2) + M(n) ± M(n) = time to merge solutions to the two subproblems ± Only need to merge two strips on either side of vertical split ± Naïve Solutions: M(n) = O(n 2 ) ± Sort the points by y-coordinate: M(n) = O(nlogn) ± Global sorting at the start: M(n) = O(n) ± Lower Bound : O(nlogn) time ± Randomized Algorithm : O(n) time [ Rabin ] 4/1/10 COT 6936 3 Post Of fi ce Problem ± Preprocess : Given set S of points in the plane representing post offices . ± Input : Query point p . ± Output : Report the closest post office to p . 4/1/10
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2 COT 6936 4 1-d Post Of fi ce Problem ± Preprocessing : Build balanced BST on S . ± O(nlogn) ± Alternatively, build a sorted array on S . ± Query Algorithm : Given a value p , identify the smallest value larger than p and the largest value smaller than p and among the two pick the one that is closest to p . ± O(log n) 4/1/10 COT 6936 5 2-d L ± Post Of fi ce Problem ± L p = ((|a x -b x |) p + (|a y -b y |) p ) 1/p ± L 2 = Euclidean distance ± L ± = max {|a x -b x |, |a y -b y |} ± Preprocessing : Build Range Tree on S . ± O(nlogn) ± Query Algorithm : Given a value p , identify the closest point to the right of p and the closest point to the left of p and among the two pick the one that is closest to p . ± O(log n) 4/1/10 COT 6936 6 2-D Range Tree ± Build the X-Tree , a balanced binary search tree on set S using the x-coordinates of the points.
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