Lecture 34 CSE 331 Nov 26, 2012

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

A property of Euclidean distance yj 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

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

Recursively find closest pairs Q R δ = min (blue, green)

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

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

Life is not so easy though Q R δ = min (blue, green)

Today’s agenda Taking care of life’s unfairness

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

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

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

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)

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

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

Greedy Algorithms Natural algorithms Reduced exponential running time to polynomial

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

A new algorithmic technique Dynamic Programming

Dynamic programming vs. Divide & Conquer

Same same because Both design recursive algorithms

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

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

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

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

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