Let yi be the y coordinate of qi for i1 to r ji 1

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Unformatted text preview: ¢ Suffices to look at points in the 2k-width strip around the median line k2 k1 Divide & Conquer Scanning the strip ā€¢ Sort all points in the strip by their y-coordinates, forming q1ā€¦qr, r n. ā€¢ Let yi be the y-coordinate of qi ā€¢ For i=1 to r ā€“ j=i-1 ā€“ While yi-yj < d ā€¢ Check the pair qi,qj ā€¢ j:=j-1 d Analysis - Analysis Divide & Conquer ā€¢ Correctness: easy ā€¢ Running time is more involved ā€¢ Can we have many qjā€™s that are within distance k from qi ? ā€¢ No ā€¢ Proof by packing argument October 30, 2003 Lecture 17: Closest Pair k 6 Divide & Conquer - Analysis Analysis, ctd. Theorem: there are at most 7 qjā€™s such that yi-yj k. Proof: ā€¢ Each such qj must lie either in the left or in the right k! k square ā€¢ Within each square, all points have distance distance k from others ā€¢ We can pack at most 4 such points into one square, so we have 8 points total (incl. qi) qi Divide & Conquer - Analysis Packing bound ā€¢ Proving ā€œ4ā€ is not obvious ā€¢ Wi l l p r o v e ā€œ 5 ā€ ā€“ Draw a disk of radius k/2 around each point ā€“ Disks are disjoint ā€“ The disk-square intersection has area (k/2)2/4 = /16 k2 ā€“ The square has area k2 ā€“ Can pack at most 16/ 5.1 points Divide & Conquer - Analysis Running time ā€¢ Divide: O(n) ā€¢ Combine: O(n log n) because we sort by y ā€¢ However, we can: ā€“ Sort all points by y at the beginning ā€“ Divide preserves the y-order of points Then combine takes only O(n) ā€¢ We get T(n)=2T(n/2)+O(n), so T(n)=O(n log n)...
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This note was uploaded on 01/15/2012 for the course COT 5405 taught by Professor Ungor during the Fall '08 term at University of Florida.

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