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Closest-Pair AlgorithmClosest-Pair Algorithm

Given: A set of points in 2-D

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Closest-Pair AlgorithmClosest-Pair Algorithm

Step 1: Sort the points in one D

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Lets sort based on the X-axis

O(n log n)

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Closest-Pair AlgorithmClosest-Pair Algorithm

8/10/2019 Lecture 37 Closest Pair

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Step 2: Split the points& i'e'&Dra a line at the id-point beteen " and#

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Closest-Pair AlgorithmClosest-Pair Algorithm

S*b-+roble

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S*b-+roble

2

8/10/2019 Lecture 37 Closest Pair

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Advantage: ,orall& e.d ha/e to0opare ea0h of the 14 points ith e/erother point'

(n-1)n2 13142 91 comparisons

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Closest-Pair AlgorithmClosest-Pair Algorithm

S*b-+roble

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S*b-+roble

2

8/10/2019 Lecture 37 Closest Pair

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Advantage: ,o& e ha/e to s*b-

probles of half the sie' 5h*s& e ha/e todo !"2 0oparisons ti0e& hi0h is 420oparisons

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Closest-Pair AlgorithmClosest-Pair Algorithm

d1d2

S*b-+roble

1

S*b-+roble

2

sol*tion d in(d1& d2)

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Advantage: 6ith 7*st one split e 0*t the

n*ber of 0oparisons in half' Ob/io*sl&e gain an e/en greater ad/antage if esplit the s*b-probles'

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Closest-Pair AlgorithmClosest-Pair Algorithm

d1d2

S*b-+roble

1

S*b-+roble

2

d in(d1& d2)

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Problem: 8oe/er& hat if the 0losest to

points are ea0h fro di9erent s*b-probles

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Closest-Pair AlgorithmClosest-Pair Algorithm

d1d2

S*b-+roble

1

S*b-+roble

2

8/10/2019 Lecture 37 Closest Pair

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8ere is an exaple here e ha/e to

0opare points fro s*b-proble 1 to thepoints in s*b-proble 2'

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Closest-Pair AlgorithmClosest-Pair Algorithm

d1d2

S*b-+roble

1

S*b-+roble

2

8/10/2019 Lecture 37 Closest Pair

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8oe/er& e onl ha/e to 0opare points

inside the folloing ;strip'<

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Closest-Pair AlgorithmClosest-Pair Algorithm

d1d2

S*b-+roble

1

S*b-+roble

2

dd

d in(d1& d2)

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Closest Pair D & CClosest Pair D & C

AlgorithmAlgorithm 1) 6e sort all points a00ording to x 0oordinates'1) 6e sort all points a00ording to x 0oordinates'

2) Di/ide all points in to hal/es'2) Di/ide all points in to hal/es'

3) >e0*rsi/el ?nd the sallest distan0es in both s*barras'3) >e0*rsi/el ?nd the sallest distan0es in both s*barras'

4) 5a@e the ini* of to sallest distan0es' Let the4) 5a@e the ini* of to sallest distan0es' Let the

ini* be d'ini* be d'

) reate an arra stripBC that stores all points hi0h are at) reate an arra stripBC that stores all points hi0h are at

ost d distan0e aa fro the iddle line di/iding the toost d distan0e aa fro the iddle line di/iding the to

sets'sets' !) ind the sallest distan0e in stripBC'!) ind the sallest distan0e in stripBC'

") >et*rn the ini* of d and the sallest distan0e") >et*rn the ini* of d and the sallest distan0e

0al0*lated in abo/e step !0al0*lated in abo/e step !

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Step 3: E*t& e 0an 0ontin*e the

ad/antage b splitting the s*b-probles'

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Closest-Pair AlgorithmClosest-Pair Algorithm

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Step 3: Fn fa0t e 0an 0ontin*e to split *ntil

ea0h s*b-proble is tri/ial& i'e'& ta@es one0oparison'

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Closest-Pair AlgorithmClosest-Pair Algorithm

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inall: 5he sol*tion to ea0h s*b-proble is

0obined *ntil the ?nal sol*tion isobtained

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Closest-Pair AlgorithmClosest-Pair Algorithm

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inall: On the last step the Gstrip. ill li@el

be /er sall' 5h*s& 0obining the tolargest s*b-probles on.t reH*ire *0hor@'

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Closest-Pair AlgorithmClosest-Pair Algorithm