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Divideand

Conquer

ClosestPairII

DesignandAnalysisofAlgorithmsI

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CorrectnessClaim

Room has one queen bed with a shared hallway bath. sink is in the bedroom, bath and shower independent

in the hallway.

Desk. Voice mail. Wireless Internet access (surcharge). Dial-up Internet access (surcharge). Cable/satellite

TV. Premium TV channels. Hair dryer. Iron/ironing board. Ceiling fan. Window opens.

Claim:Let beasplitpairwithd(p,q)

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ProofofCorrectnessClaim(A)Let

Note:Since

Proofof(A)[pandqaremembersofSyi.e.]

Note:

=>

StripthatdefinesSy

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ProofofCorrectnessClaim(B)(B):p=(x1,y1)andq=(x2,y2)areatmost7posiHons

apartinSy

KeyPicture:draw boxeswithcenterand

boommin{y1,y2}

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ProofofCorrectnessClaim(B)Lemma1:allpointsofSywith

y-coordinatebetweenthoseof

pandq,inclusive,lieinoneof

these8boxes.

Proof:First,recally-coordinatesofp,qdifferby

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ProofofCorrectnessClaim(B)Lemma2:Atmostonepoint

ofPineachbox.

Proof:bycontradicHonSupposea,blieinthesamebox.Then:

I. a,bareeitherbothinQorbothinRII. But(i)and(ii)contradictthedefiniHonof

(assmallestdistancebetweenpairsofpoints

inQorinR)

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FinalWrap-Up

Lemmas1and2=>atmost8

pointsinthispicture(includingpandq)

=>PosiHonsofp,qinSydiffer

byatmost7