1. Lecture 31
CSE 331
Nov 11, 2011

2. HW 8 due today I will not take any HW after 1:15pm
Q1 and Q 2 in separate piles
I will not take any HW after 1:15pm

3. Other HW related stuff Solutions to HW 8 on Monday
HW 7 should be available for pickup from Monday
HW 9 has been posted on the blog

4. Review Session Details

5. Optimal MST algorithms

6. Need 2 blog volunteers

7. Counting Inversions Input: n distinct numbers a1,a2,…,an
Inversion: (i,j) with i < j s.t. ai > aj
Output: Number of inversions

8. 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

9. Three kinds of inversion10 7 21 20
100 1
Non-crossing inversions are counted recursively

10. Today’s agenda
Divide and Conquer algorithm for counting the # inversions

11. HW 8 due today I will not take any HW after 1:15pm
Q1 and Q 2 in separate piles
I will not take any HW after 1:15pm

12. Mergesort-Count algorithm
Input: a1, a2, …, an
Output: Numbers in sorted order+ #inversion
T(2) = c
T(n) = 2T(n/2) + cn
MergeSortCount( a, n )
If n = 1 return ( 0 , a1)
If n = 2 return ( a1 > a2, min(a1,a2); max(a1,a2))
O(n log n) time
aL = a1,…, an/2
aR = an/2+1,…, an
(cL, aL) = MergeSortCount(aL, n/2)
(cR, aR) = MergeSortCount(aR, n/2)
O(n)
Counts #crossing-inversions+ MERGE
(c, a) = MERGE-COUNT(aL,aR)
return (c+cL+cR,a)

13. Rest of today’s agenda
MERGE-COUNT
Computing closest pair of points

14. Closest pairs of points
Input: n 2-D points P = {p1,…,pn}; pi=(xi,yi)
d(pi,pj) = ( (xi-xj)2+(yi-yj)2)1/2
Output: Points p and q that are closest

15. Group Talk time
O(n2) time algorithm?
1-D problem in time O(n log n) ?

16. Sorting to rescue in 2-D?
Pick pairs of points closest in x co-ordinate
Pick pairs of points closest in y co-ordinate
Choose the better of the two

17. Rest of today’s agenda
Divide and Conquer based algorithm