Ham Sandwich Theorem Carola Wenk 3/8/16 1 CMPS 6640/4040 - - PowerPoint PPT Presentation

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CMPS 6640/4040 Computational Geometry Spring 2016 Ham Sandwich Theorem Carola Wenk 3/8/16 1 CMPS 6640/4040 Computational Geometry Ham-Sandwich Theorem Theorem: Let P and Q be two finite point sets in the plane Then there exists a line l such

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3/8/16 CMPS 6640/4040 Computational Geometry 1

CMPS 6640/4040 Computational Geometry Spring 2016

Ham Sandwich Theorem

Carola Wenk

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3/8/16 CMPS 6640/4040 Computational Geometry 2

Ham-Sandwich Theorem

Theorem: Let P and Q be two finite point sets in the plane Then there exists a line l such that on each side of l there are at most |P|/2 points of P and at most |Q|/2 points of Q.

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3/8/16 CMPS 6640/4040 Computational Geometry 3

Ham-Sandwich Theorem

Proof: Find a line l such that on each side of l there are at most |P|/2 points of P. Then rotate l counter-clockwise in such a way that there are always at most |P|/2 points of P on each side of l.

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3/8/16 CMPS 6640/4040 Computational Geometry 4

Rotation

Left: 4 Right: 4

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3/8/16 CMPS 6640/4040 Computational Geometry 5

Rotation

Left: 4 Right: 3 Rotate around this point now

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3/8/16 CMPS 6640/4040 Computational Geometry 6

Rotation

Left: 4 Right: 4

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3/8/16 CMPS 6640/4040 Computational Geometry 7

Rotation

Left: 3 Right: 4 rotate

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3/8/16 CMPS 6640/4040 Computational Geometry 8

Rotation

Left: 3 Right: 4 rotate

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3/8/16 CMPS 6640/4040 Computational Geometry 9

Rotation

Left: 4 Right: 3 rotate

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3/8/16 CMPS 6640/4040 Computational Geometry 10

Rotation

Left: 3 Right: 4 rotate

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3/8/16 CMPS 6640/4040 Computational Geometry 11

Rotation

Left: 4 Right: 3 rotate

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3/8/16 CMPS 6640/4040 Computational Geometry 12

Rotation

Left: 2 Right: 4 Rotate around this point now

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3/8/16 CMPS 6640/4040 Computational Geometry 13

Rotation

Left: 4 Right: 4 Rotate around this point now

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3/8/16 CMPS 6640/4040 Computational Geometry 14

Rotation

Left: 4 Right: 3 rotate

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3/8/16 CMPS 6640/4040 Computational Geometry 15

Rotation

Left: 3 Right: 4 rotate

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3/8/16 CMPS 6640/4040 Computational Geometry 16

Rotation

Left: 4 Right: 3 rotate

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3/8/16 CMPS 6640/4040 Computational Geometry 17

Rotation

Left: 4 Right: 4

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3/8/16 CMPS 6640/4040 Computational Geometry 18

Proof Continued

In general, choose the rotation point such that the number of points on each side of l does not change.

k points m points k points m points rotate rotate

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3/8/16 CMPS 6640/4040 Computational Geometry 19

Proof Continued

Throughout the rotation, there are at most |P|/2 points on each side of l. After 180 rotation, we get the line which is l but directed in the other direction. Let t be the number of blue points to the left of l at the

beginning. In the end, t points are on the right side of l, so

|Q|-t-1 are on the left side. Therefore, there must have been

ne orientation of l such that there were t most |Q|/2 points
n each side of l.