1. Lecture 34
CSE 331
Nov 26, 2012

2. 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

3. A property of Euclidean distanceyj
d(pi,pj) = ( (xi-xj)2+(yi-yj)2)1/2 yi xi xj
The distance is larger than the x or y-coord difference

4. Dividing up P Q R
First n/2 points according to the x-coord

5. Recursively find closest pairsQ R
δ = min (blue, green)

6. An aside: maintain sorted lists
Px and Py are P sorted by x-coord and y-coord
Qx, Qy, Rx, Ry can be computed from Px and Py in O(n) time

7. An easy case R > δ Q All “crossing” pairs have distance > δ
δ = min (blue, green)

8. Life is not so easy thoughQ R
δ = min (blue, green)

9. Today’s agenda
Taking care of life’s unfairness

10. Euclid to the rescue (?) d(pi,pj) = ( (xi-xj)2+(yi-yj)2)1/2 yj yi xi
The distance is larger than the x or y-coord difference

11. Life is not so easy thoughδ Q R
> δ
> δ
> δ
δ = min (blue, green)

12. All we have to do now δ R Q S
Figure if a pair in S has distance < δ S
δ = min (blue, green)

13. Assume can be done in O(n)
The algorithm so far…
O(n log n) + T(n)
Input: n 2-D points P = {p1,…,pn}; pi=(xi,yi)
Sort P to get Px and Py
O(n log n)
T(< 4) = c
Closest-Pair (Px, Py)
T(n) = 2T(n/2) + cn
If n < 4 then find closest point by brute-force
Q is first half of Px and R is the rest
O(n)
Compute Qx, Qy, Rx and Ry
O(n)
(q0,q1) = Closest-Pair (Qx, Qy)
O(n log n) overall
(r0,r1) = Closest-Pair (Rx, Ry)
O(n)
δ = min ( d(q0,q1), d(r0,r1) )
O(n)
S = points (x,y) in P s.t. |x – x*| < δ
return Closest-in-box (S, (q0,q1), (r0,r1))
Assume can be done in O(n)

14. Rest of today’s agenda
Implement Closest-in-box in O(n) time

15. Three general techniques
High level view of CSE 331
Problem Statement
Problem Definition
Three general techniques
Algorithm
“Implementation”
Data Structures
Analysis
Correctness+Runtime Analysis

16. Greedy Algorithms Natural algorithms
Reduced exponential running time to polynomial

17. Divide and Conquer Recursive algorithmic paradigm
Reduced large polynomial time to smaller polynomial time

18. A new algorithmic technique
Dynamic Programming

19. Dynamic programming vs. Divide & Conquer

20. Same same because
Both design recursive algorithms

21. Different because
Dynamic programming is smarter about solving recursive sub-problems

22. Using Memory to be smarter
Pow (a,n)
// n is even and ≥ 2
return t * t
t= Pow(a,n/2)
Pow (a,n)
// n is even and ≥ 2
return Pow(a,n/2) * Pow(a, n/2)
O(n) as we recompute!
O(log n) as we compute only once

23. End of Semester blues
Can only do one thing at any day: what is the optimal schedule to obtain maximum value?
Write up a term paper
(30)
(3)
(2)
(5)
(10)
Party!
Exam study
331 HW
Project
Monday
Tuesday
Wednesday
Thursday
Friday

24. Previous Greedy algorithm
Order by end time and pick jobs greedily
Greedy value = 5+2+3= 10
Write up a term paper
(10)
OPT = 30
Party!
(2)
Exam study
(5)
331 HW
(3)
Project
(30)
Monday
Tuesday
Wednesday
Thursday
Friday

25. Today’s agenda Formal definition of the problem
Start designing a recursive algorithm for the problem