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A Simple 3-Approximation of Minimum Manhattan Networks Bernhard Fuchs and Anna Schulze TU Braunschweig and Uni K oln CTW 2008 B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 1 / 14 Manhattan networks Given a set of

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SLIDE 1

A Simple 3-Approximation of Minimum Manhattan Networks

Bernhard Fuchs and Anna Schulze

TU Braunschweig and Uni K¨

  • ln

CTW 2008

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 1 / 14

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SLIDE 2

Manhattan networks

Given a set of points in the plane,

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 2 / 14

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SLIDE 3

Manhattan networks

Given a set of points in the plane, a Manhattan network contains all pairwise shortest rectilinear paths.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 2 / 14

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SLIDE 4

Manhattan networks

Given a set of points in the plane, a Manhattan network contains all pairwise shortest rectilinear paths.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 2 / 14

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SLIDE 5

Manhattan networks

Given a set of points in the plane, a Manhattan network contains all pairwise shortest rectilinear paths. Task: Find a Minimum Manhattan Network (MMN)!

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 2 / 14

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SLIDE 6

Manhattan networks

Given a set of points in the plane, a Manhattan network contains all pairwise shortest rectilinear paths. Task: Find a Minimum Manhattan Network (MMN)!

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 2 / 14

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SLIDE 7

Simple heuristics – Hannan grid

First heuristic: Take the whole Hannan grid, length Ω(n2).

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 3 / 14

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SLIDE 8

Simple heuristics – Hannan grid

First heuristic: Take the whole Hannan grid, length Ω(n2).

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 3 / 14

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SLIDE 9

Simple heuristics – Hannan grid

First heuristic: Take the whole Hannan grid, length Ω(n2). MMN has length O(n).

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 3 / 14

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SLIDE 10

Simple heuristics – Hannan grid

First heuristic: Take the whole Hannan grid, length Ω(n2). MMN has length O(n).

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 3 / 14

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SLIDE 11

Simple heuristics – Hannan grid

First heuristic: Take the whole Hannan grid, length Ω(n2). MMN has length O(n).

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 3 / 14

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SLIDE 12

Simple heuristics – Hannan grid

First heuristic: Take the whole Hannan grid, length Ω(n2). MMN has length O(n). No constant factor approximation.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 3 / 14

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SLIDE 13

Simple heuristics – Hannan grid

First heuristic: Take the whole Hannan grid, length Ω(n2). MMN has length O(n). No constant factor approximation. Next idea: Just insert critical rectangles!

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 3 / 14

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SLIDE 14

Critical rectangles

A critical rectangle contains exactly the two points spanning it.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 4 / 14

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SLIDE 15

Critical rectangles

A critical rectangle contains exactly the two points spanning it.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 4 / 14

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SLIDE 16

Critical rectangles

A critical rectangle contains exactly the two points spanning it. It obviously suffices to consider critical rectangles.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 4 / 14

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SLIDE 17

Critical rectangles

A critical rectangle contains exactly the two points spanning it. It obviously suffices to consider critical rectangles. Note: Each Manhattan network has to cross such a rectangle!

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 4 / 14

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SLIDE 18

Critical rectangles

A critical rectangle contains exactly the two points spanning it. It obviously suffices to consider critical rectangles. Note: Each Manhattan network has to cross such a rectangle!

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 4 / 14

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SLIDE 19

Critical rectangles

A critical rectangle contains exactly the two points spanning it. It obviously suffices to consider critical rectangles. Note: Each Manhattan network has to cross such a rectangle!

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 4 / 14

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SLIDE 20

Critical rectangles

A critical rectangle contains exactly the two points spanning it. It obviously suffices to consider critical rectangles. Note: Each Manhattan network has to cross such a rectangle!

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 4 / 14

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SLIDE 21

Too many critical rectangles

Add one more point.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 5 / 14

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SLIDE 22

Too many critical rectangles

Add one more point. Many more critical rectangles, total length Ω(n2).

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 5 / 14

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SLIDE 23

Too many critical rectangles

Add one more point. Many more critical rectangles, total length Ω(n2). MMN is basically binary tree, length O(n log n).

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 5 / 14

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SLIDE 24

Too many critical rectangles

Add one more point. Many more critical rectangles, total length Ω(n2). MMN is basically binary tree, length O(n log n).

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 5 / 14

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SLIDE 25

Too many critical rectangles

Add one more point. Many more critical rectangles, total length Ω(n2). MMN is basically binary tree, length O(n log n). Still no constant factor approximation.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 5 / 14

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SLIDE 26

Prior work

Complexity of MMN: Still unknown!

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 6 / 14

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SLIDE 27

Prior work

Complexity of MMN: Still unknown! Approximation algorithms so far:

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 6 / 14

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SLIDE 28

Prior work

Complexity of MMN: Still unknown! Approximation algorithms so far: [Gudmunsson et al.] 99 8-approx. in O(n3) 4-approx. in O(n log n)

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 6 / 14

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SLIDE 29

Prior work

Complexity of MMN: Still unknown! Approximation algorithms so far: [Gudmunsson et al.] 99 8-approx. in O(n3) 4-approx. in O(n log n) [Kato et al.] 02 2-approx. in O(n3)

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 6 / 14

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SLIDE 30

Prior work

Complexity of MMN: Still unknown! Approximation algorithms so far: [Gudmunsson et al.] 99 8-approx. in O(n3) 4-approx. in O(n log n) [Kato et al.] 02 2-approx. in O(n3) incomplete

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 6 / 14

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SLIDE 31

Prior work

Complexity of MMN: Still unknown! Approximation algorithms so far: [Gudmunsson et al.] 99 8-approx. in O(n3) 4-approx. in O(n log n) [Kato et al.] 02 2-approx. in O(n3) incomplete [Benkert et al.] 04 3-approx. in O(n log n)

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 6 / 14

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SLIDE 32

Prior work

Complexity of MMN: Still unknown! Approximation algorithms so far: [Gudmunsson et al.] 99 8-approx. in O(n3) 4-approx. in O(n log n) [Kato et al.] 02 2-approx. in O(n3) incomplete [Benkert et al.] 04 3-approx. in O(n log n) [Chepoi et al.] 05 2-approx. via LP-rounding

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 6 / 14

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SLIDE 33

Prior work

Complexity of MMN: Still unknown! Approximation algorithms so far: [Gudmunsson et al.] 99 8-approx. in O(n3) 4-approx. in O(n log n) [Kato et al.] 02 2-approx. in O(n3) incomplete [Benkert et al.] 04 3-approx. in O(n log n) [Chepoi et al.] 05 2-approx. via LP-rounding [Seibert, Unger] 05 1.5-approx. in O(n3)

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 6 / 14

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SLIDE 34

Prior work

Complexity of MMN: Still unknown! Approximation algorithms so far: [Gudmunsson et al.] 99 8-approx. in O(n3) 4-approx. in O(n log n) [Kato et al.] 02 2-approx. in O(n3) incomplete [Benkert et al.] 04 3-approx. in O(n log n) [Chepoi et al.] 05 2-approx. via LP-rounding [Seibert, Unger] 05 1.5-approx. in O(n3) incomplete

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 6 / 14

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SLIDE 35

Prior work

Complexity of MMN: Still unknown! Approximation algorithms so far: [Gudmunsson et al.] 99 8-approx. in O(n3) 4-approx. in O(n log n) [Kato et al.] 02 2-approx. in O(n3) incomplete [Benkert et al.] 04 3-approx. in O(n log n) [Chepoi et al.] 05 2-approx. via LP-rounding [Seibert, Unger] 05 1.5-approx. in O(n3) incomplete [Nouioua] — 2-approx. in O(n log n)

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 6 / 14

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SLIDE 36

Prior work

Complexity of MMN: Still unknown! Approximation algorithms so far: [Gudmunsson et al.] 99 8-approx. in O(n3) 4-approx. in O(n log n) [Kato et al.] 02 2-approx. in O(n3) incomplete [Benkert et al.] 04 3-approx. in O(n log n) [Chepoi et al.] 05 2-approx. via LP-rounding [Seibert, Unger] 05 1.5-approx. in O(n3) incomplete [Nouioua] — 2-approx. in O(n log n) unpublished

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 6 / 14

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SLIDE 37

Prior work

Complexity of MMN: Still unknown! Approximation algorithms so far: [Gudmunsson et al.] 99 8-approx. in O(n3) 4-approx. in O(n log n) [Kato et al.] 02 2-approx. in O(n3) incomplete [Benkert et al.] 04 3-approx. in O(n log n) [Chepoi et al.] 05 2-approx. via LP-rounding [Seibert, Unger] 05 1.5-approx. in O(n3) incomplete [Nouioua] — 2-approx. in O(n log n) unpublished

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 6 / 14

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SLIDE 38

Prior work

Complexity of MMN: Still unknown! Approximation algorithms so far: [Gudmunsson et al.] 99 8-approx. in O(n3) 4-approx. in O(n log n) [Kato et al.] 02 2-approx. in O(n3) incomplete [Benkert et al.] 04 3-approx. in O(n log n) [Chepoi et al.] 05 2-approx. via LP-rounding [Seibert, Unger] 05 1.5-approx. in O(n3) incomplete [Nouioua] — 2-approx. in O(n log n) unpublished Our algorithm: A new 3-approximation in O(n log n).

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 6 / 14

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SLIDE 39

Prior work

Complexity of MMN: Still unknown! Approximation algorithms so far: [Gudmunsson et al.] 99 8-approx. in O(n3) 4-approx. in O(n log n) [Kato et al.] 02 2-approx. in O(n3) incomplete [Benkert et al.] 04 3-approx. in O(n log n) [Chepoi et al.] 05 2-approx. via LP-rounding [Seibert, Unger] 05 1.5-approx. in O(n3) incomplete [Nouioua] — 2-approx. in O(n log n) unpublished Our algorithm: A new 3-approximation in O(n log n). Much simpler than [Benkert et al.], both algorithm and proof.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 6 / 14

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SLIDE 40

Algorithm outline

Our algorithm has two phases:

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 7 / 14

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SLIDE 41

Algorithm outline

Our algorithm has two phases:

Phase I

A horizontal and a vertical sweep adding line segments ‘on-the-fly’.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 7 / 14

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SLIDE 42

Algorithm outline

Our algorithm has two phases:

Phase I

A horizontal and a vertical sweep adding line segments ‘on-the-fly’.

Phase II

A standard 2-approximation algorithm inside so-called ‘staircases’.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 7 / 14

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SLIDE 43

Phase I - The sweep

Consider critical rectangles spanned by horizontal, or x-neighbors.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 8 / 14

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SLIDE 44

Phase I - The sweep

Consider critical rectangles spanned by horizontal, or x-neighbors.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 8 / 14

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SLIDE 45

Phase I - The sweep

Consider critical rectangles spanned by horizontal, or x-neighbors. Add vertical sides of rectangle.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 8 / 14

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SLIDE 46

Phase I - The sweep

Consider critical rectangles spanned by horizontal, or x-neighbors. Add vertical sides of rectangle. Iterate through rectangles from left to right.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 8 / 14

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SLIDE 47

Phase I - The sweep

Consider critical rectangles spanned by horizontal, or x-neighbors. Add vertical sides of rectangle. Iterate through rectangles from left to right.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 8 / 14

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SLIDE 48

Phase I - The sweep

Consider critical rectangles spanned by horizontal, or x-neighbors. Add vertical sides of rectangle. Iterate through rectangles from left to right.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 8 / 14

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SLIDE 49

Phase I - The sweep

Consider critical rectangles spanned by horizontal, or x-neighbors. Add vertical sides of rectangle. Iterate through rectangles from left to right.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 8 / 14

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SLIDE 50

Phase I - The sweep

Consider critical rectangles spanned by horizontal, or x-neighbors. Add vertical sides of rectangle. Iterate through rectangles from left to right.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 8 / 14

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SLIDE 51

Phase I - The sweep

Consider critical rectangles spanned by horizontal, or x-neighbors. Add vertical sides of rectangle. Iterate through rectangles from left to right.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 8 / 14

slide-52
SLIDE 52

Phase I - The sweep

Consider critical rectangles spanned by horizontal, or x-neighbors. Add vertical sides of rectangle. Iterate through rectangles from left to right.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 8 / 14

slide-53
SLIDE 53

Phase I - The sweep

Consider critical rectangles spanned by horizontal, or x-neighbors. Add vertical sides of rectangle. Iterate through rectangles from left to right.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 8 / 14

slide-54
SLIDE 54

Phase I - The sweep

Consider critical rectangles spanned by horizontal, or x-neighbors. Add vertical sides of rectangle. Iterate through rectangles from left to right.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 8 / 14

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SLIDE 55

Phase I - The sweep

Proceed likewise with y-neighbors.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 8 / 14

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SLIDE 56

Phase I - The sweep

Proceed likewise with y-neighbors.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 8 / 14

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SLIDE 57

Phase I - The sweep

Proceed likewise with y-neighbors.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 8 / 14

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SLIDE 58

Phase I - The sweep

Proceed likewise with y-neighbors.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 8 / 14

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SLIDE 59

Phase I - The sweep

Proceed likewise with y-neighbors.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 8 / 14

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SLIDE 60

Phase I - The sweep

Proceed likewise with y-neighbors.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 8 / 14

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SLIDE 61

Phase I - The sweep

Proceed likewise with y-neighbors.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 8 / 14

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SLIDE 62

Phase I - The sweep

Proceed likewise with y-neighbors.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 8 / 14

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SLIDE 63

Phase I - The sweep

Proceed likewise with y-neighbors.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 8 / 14

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SLIDE 64

Phase I - The sweep

Proceed likewise with y-neighbors.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 8 / 14

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SLIDE 65

Phase I - The sweep

Proceed likewise with y-neighbors.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 8 / 14

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SLIDE 66

Phase I - The sweep

Proceed likewise with y-neighbors.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 8 / 14

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SLIDE 67

After the sweep

In general, no Manhattan network after sweep.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 9 / 14

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SLIDE 68

After the sweep

In general, no Manhattan network after sweep.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 9 / 14

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SLIDE 69

After the sweep

In general, no Manhattan network after sweep.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 9 / 14

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SLIDE 70

After the sweep

In general, no Manhattan network after sweep.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 9 / 14

slide-71
SLIDE 71

After the sweep

In general, no Manhattan network after sweep.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 9 / 14

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SLIDE 72

After the sweep

In general, no Manhattan network after sweep.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 9 / 14

slide-73
SLIDE 73

After the sweep

In general, no Manhattan network after sweep.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 9 / 14

slide-74
SLIDE 74

After the sweep

In general, no Manhattan network after sweep.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 9 / 14

slide-75
SLIDE 75

After the sweep

In general, no Manhattan network after sweep.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 9 / 14

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SLIDE 76

After the sweep

In general, no Manhattan network after sweep.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 9 / 14

slide-77
SLIDE 77

After the sweep

In general, no Manhattan network after sweep. So-called staircases still empty.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 9 / 14

slide-78
SLIDE 78

After the sweep

A

In general, no Manhattan network after sweep. So-called staircases still empty. Call A the staircase area.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 9 / 14

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SLIDE 79

After the sweep

Definition (staircase)

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 10 / 14

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SLIDE 80

After the sweep

v1 v2 v3 v4

Definition (staircase)

k sequence points (v1, . . . , vk).

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 10 / 14

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SLIDE 81

After the sweep

v1 v2 v3 v4 by bx

Definition (staircase)

k sequence points (v1, . . . , vk). Two base points bx, by. (bx = by possible.)

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 10 / 14

slide-82
SLIDE 82

After the sweep

v1 v2 v3 v4 by bx

Definition (staircase)

k sequence points (v1, . . . , vk). Two base points bx, by. (bx = by possible.) For all vi, bx is the x-neighbor of vi in the third quadrant of vi.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 10 / 14

slide-83
SLIDE 83

After the sweep

v1 v2 v3 v4 by bx

Definition (staircase)

k sequence points (v1, . . . , vk). Two base points bx, by. (bx = by possible.) For all vi, bx is the x-neighbor of vi in the third quadrant of vi.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 10 / 14

slide-84
SLIDE 84

After the sweep

v1 v2 v3 v4 by bx

Definition (staircase)

k sequence points (v1, . . . , vk). Two base points bx, by. (bx = by possible.) For all vi, bx is the x-neighbor of vi in the third quadrant of vi.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 10 / 14

slide-85
SLIDE 85

After the sweep

v1 v2 v3 v4 by bx

Definition (staircase)

k sequence points (v1, . . . , vk). Two base points bx, by. (bx = by possible.) For all vi, bx is the x-neighbor of vi in the third quadrant of vi.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 10 / 14

slide-86
SLIDE 86

After the sweep

v1 v2 v3 v4 by bx

Definition (staircase)

k sequence points (v1, . . . , vk). Two base points bx, by. (bx = by possible.) For all vi, bx is the x-neighbor of vi in the third quadrant of vi.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 10 / 14

slide-87
SLIDE 87

After the sweep

v1 v2 v3 v4 by bx

Definition (staircase)

k sequence points (v1, . . . , vk). Two base points bx, by. (bx = by possible.) For all vi, bx is the x-neighbor of vi in the third quadrant of vi.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 10 / 14

slide-88
SLIDE 88

After the sweep

v1 v2 v3 v4 by bx

Definition (staircase)

k sequence points (v1, . . . , vk). Two base points bx, by. (bx = by possible.) For all vi, bx is the x-neighbor of vi in the third quadrant of vi. For all vi, by is the y-neighbor of vi in the third quadrant of vi.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 10 / 14

slide-89
SLIDE 89

After the sweep

v1 v2 v3 v4 by bx

Observation

The grey shaded areas contain no points.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 10 / 14

slide-90
SLIDE 90

After the sweep

v1 v2 v3 v4 by bx c A

Observation

The grey shaded areas contain no points. No sweep lines lie inside the staircase area A.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 10 / 14

slide-91
SLIDE 91

After the sweep

v1 v2 v3 v4 by bx c A

Observation

The grey shaded areas contain no points. No sweep lines lie inside the staircase area A. ⇒ Points v3, . . . , vk−2 need to be connected to cross point c.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 10 / 14

slide-92
SLIDE 92

After the sweep

Lemma

Except for staircase areas, all critical pairs of points are connected via shortest paths after the sweep.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 11 / 14

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SLIDE 93

After the sweep

A1

A2 A3 A4

Lemma

Except for staircase areas, all critical pairs of points are connected via shortest paths after the sweep.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 11 / 14

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SLIDE 94

After the sweep

A1

A2 A3 A4

Lemma

Except for staircase areas, all critical pairs of points are connected via shortest paths after the sweep. The staircase areas Ai are bordered by as many line segments from the sweep step as possible,

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 11 / 14

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SLIDE 95

After the sweep

A1

A2 A3 A4

Lemma

Except for staircase areas, all critical pairs of points are connected via shortest paths after the sweep. The staircase areas Ai are bordered by as many line segments from the sweep step as possible, and are as small as possible.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 11 / 14

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SLIDE 96

Approximation analysis

Consider the approximation seperately: inside staircases (A =

i Ai, Phase II), and

  • utside staircases (A := R2 \ A, Phase I).
  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 12 / 14

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SLIDE 97

Approximation analysis

Consider the approximation seperately: inside staircases (A =

i Ai, Phase II), and

  • utside staircases (A := R2 \ A, Phase I).

Phase I (Area A)

By construction, one of the two lines inserted by the sweep is justified.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 12 / 14

slide-98
SLIDE 98

Approximation analysis

Consider the approximation seperately: inside staircases (A =

i Ai, Phase II), and

  • utside staircases (A := R2 \ A, Phase I).

Phase I (Area A)

By construction, one of the two lines inserted by the sweep is justified. ⇒ 2-Approximation.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 12 / 14

slide-99
SLIDE 99

Approximation analysis

Consider the approximation seperately: inside staircases (A =

i Ai, Phase II), and

  • utside staircases (A := R2 \ A, Phase I).

Phase I (Area A)

By construction, one of the two lines inserted by the sweep is justified. ⇒ 2-Approximation?! Unfortunately not:

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 12 / 14

slide-100
SLIDE 100

Approximation analysis

Consider the approximation seperately: inside staircases (A =

i Ai, Phase II), and

  • utside staircases (A := R2 \ A, Phase I).

Phase I (Area A)

By construction, one of the two lines inserted by the sweep is justified. ⇒ 2-Approximation?! Unfortunately not:

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 12 / 14

slide-101
SLIDE 101

Approximation analysis

Consider the approximation seperately: inside staircases (A =

i Ai, Phase II), and

  • utside staircases (A := R2 \ A, Phase I).

Phase I (Area A)

By construction, one of the two lines inserted by the sweep is justified. ⇒ 2-Approximation?! Unfortunately not:

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 12 / 14

slide-102
SLIDE 102

Approximation analysis

Consider the approximation seperately: inside staircases (A =

i Ai, Phase II), and

  • utside staircases (A := R2 \ A, Phase I).

Phase I (Area A)

By construction, one of the two lines inserted by the sweep is justified. ⇒ 2-Approximation?! Unfortunately not:

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 12 / 14

slide-103
SLIDE 103

Approximation analysis

Consider the approximation seperately: inside staircases (A =

i Ai, Phase II), and

  • utside staircases (A := R2 \ A, Phase I).

Phase I (Area A)

By construction, one of the two lines inserted by the sweep is justified. ⇒ 3-Approximation.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 12 / 14

slide-104
SLIDE 104

Approximation analysis

Phase II (Area A)

Use standard 2-approximation for staircases.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 13 / 14

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SLIDE 105

Approximation analysis

Phase II (Area A)

Use standard 2-approximation for staircases. Let A∗ be optimal staircase areas. Note: A ⊆ A∗.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 13 / 14

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SLIDE 106

Approximation analysis

Phase II (Area A)

Use standard 2-approximation for staircases. Let A∗ be optimal staircase areas. Note: A ⊆ A∗. Let alg = algA + algA and opt = optA∗ + optA∗.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 13 / 14

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SLIDE 107

Approximation analysis

Phase II (Area A)

Use standard 2-approximation for staircases. Let A∗ be optimal staircase areas. Note: A ⊆ A∗. Let alg = algA + algA and opt = optA∗ + optA∗. Phase I: algA ≤ 3 · optA∗.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 13 / 14

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SLIDE 108

Approximation analysis

Phase II (Area A)

Use standard 2-approximation for staircases. Let A∗ be optimal staircase areas. Note: A ⊆ A∗. Let alg = algA + algA and opt = optA∗ + optA∗. Phase I: algA ≤ 3 · optA∗. Phase II: algA ≤ 2 · optA ≤ 2 · optA∗.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 13 / 14

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SLIDE 109

Approximation analysis

Phase II (Area A)

Use standard 2-approximation for staircases. Let A∗ be optimal staircase areas. Note: A ⊆ A∗. Let alg = algA + algA and opt = optA∗ + optA∗. Phase I: algA ≤ 3 · optA∗. Phase II: algA ≤ 2 · optA ≤ 2 · optA∗. Altogether: alg = algA + algA ≤ 2 · optA∗ + 3 · optA∗ ≤ 3 · opt.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 13 / 14

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SLIDE 110

Conclusion

We presented a simplified 3-approximation algorithm for MMNs. Future work:

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 14 / 14

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SLIDE 111

Conclusion

We presented a simplified 3-approximation algorithm for MMNs. Future work: Design better approximation algorithms.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 14 / 14

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SLIDE 112

Conclusion

We presented a simplified 3-approximation algorithm for MMNs. Future work: Design better approximation algorithms. Design PTAS.

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 14 / 14

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SLIDE 113

Conclusion

We presented a simplified 3-approximation algorithm for MMNs. Future work: Design better approximation algorithms. Design PTAS. Design efficient optimal algorithm or prove NP-hardness!

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 14 / 14

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SLIDE 114

Conclusion

We presented a simplified 3-approximation algorithm for MMNs. Future work: Design better approximation algorithms. Design PTAS. Design efficient optimal algorithm or prove NP-hardness! Thank you!

  • B. Fuchs (TU Braunschweig)

Simple 3-Approximation of MMNs CTW 2008 14 / 14