Input: points p1, p2,

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Example

• Input points p1, p2,,p8 in a plane.
• Find a line L1 such that 4 points are on one
  side and the other 4 points are on the other
  side.

L1
p6
p1
p7
p2
p3
p8
p5
p4
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Consider the left four points p1,p2,p3,p4. Find
a line L2 such that 2 points are on one side and
the other 2 points are on the other side.d1
Closest-pair (Region 1) dist(p1,p2) 1. d2
Closest- pair (Region 2) dist(p3,p4)
2.5dmin(d1 ,d2) 1
L2
L1
Reg 1
Reg 2
p1
1
p7
p6
p2
p3
2.5
p8
p5
p4
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• Delete the points in Region 1 and 2 further than
  d 1 from L2
• Compare the distance dist(p1,p3) with d. Here
  dist(p1,p3) 3 gt d, d is not updated.
• Closest-pair (Region 1 and 2) dist(p1,p2) 1.

L3
Reg 1
Reg 2
L1
p1
p6
p7
3
p3
p8
p5
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• For the four points p5,p6,p7,p8, we use a line L3
  to divide them and use similar method to find
• d1 Closest-pair (Region 3) dist(p5,p6)
  2.5
• d2 Closest- pair (Region 4) dist(p7,p8)
  2
• d min(d1 ,d2) 2

L1
L3
Reg 3
Reg 4
L2
Reg 2
Reg 1
p6
p1
p7
1
2
2.5
p2
p3
p4
p8
p5
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• Delete the points in Region 1 and 2 further than
  d 2 from L3
• Compare the distance dist(p6,p7) with d.
• Here dist(p6,p7) 0.5 lt d, d is updated.
• Closest-pair (Region 3 and 4) dist(p6,p7) 0.5.

L1
L3
Reg 3
Reg 4
L2
Reg 2
Reg 1
p6
p1
p7
1
0.5
p2
p3
p4
p8
p5
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• Consider Region 1 , 2 and Region 3, 4 as two
  large regions.
• d1 Closest-pair (Region 1 and 2) dist(p1,p2)
  1
• d2 Closest- pair (Region 3 and 4) dist(p6,p7)
  0.5
• d min(d1 ,d2) 0.5

L1
L3
Reg 3
Reg 4
L2
Reg 2
Reg 1
p6
p1
p7
1
0.5
p2
p3
p4
p8
p5
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• Delete the points in Region 1, 2, 3 and 4 further
  than d 0.5 from L1
• Here only one point p4 is left, d is not updated.
• Closest-pair (Region 1, 2, 3 and 4) dist(p6,p7)
  0.5.

L1
L3
Reg 3
Reg 4
L2
Reg 2
Reg 1
p4