Lecture 32 CSE 331 Nov 18, 2009

HW 8 solutions Friday

Counting Inversions Input: n distinct numbers a 1,a 2,…,a n Inversion: (i,j) with i a j Output: Number of inversions

Divide and Conquer Divide up the problem into at least two sub-problems Recursively solve the sub-problems “Patch up” the solutions to the sub-problems for the final solution Solve the stronger problem of counting inversions + sorting

Three kinds of inversion Non-crossing inversions are counted recursively

Mergesort-Count algorithm Input: a 1, a 2, …, a n Output: Numbers in sorted order+ #inversion MergeSortCount( a, n ) If n = 2 return the order ( a1 > a2, min(a 1,a 2 ); max(a 1,a 2 )) a L = a 1,…, a n/2 a R = a n/2+1,…, a n return (c+c L +c R,a) (c L, a L ) = MergeSortCount(a L, n/2) (c R, a R ) = MergeSortCount(a R, n/2) (c, a) = MERGE-COUNT(a L,a R ) Counts #crossing-inversions+ MERGE O(n) T(2) = c T(n) = 2T(n/2) + cn O(n log n) time

Today’s agenda MERGE-COUNT Computing closest pair of points