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Approximating Minimum Manhattan Networks in Higher Dimensions - - PowerPoint PPT Presentation

Oct 29, 2022 27 likes •717 views

Approximating Minimum Manhattan Networks in Higher Dimensions Aparna Das Emden R. Gansner Michael Kaufmann Stephen Kobourov Joachim Spoerhase Alexander Wolff ESA11 Lehrstuhl f ur Informatik I Universit at W urzburg,

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SLIDE 1

Approximating Minimum Manhattan Networks in Higher Dimensions

Aparna Das · Emden R. Gansner · Michael Kaufmann Stephen Kobourov · Joachim Spoerhase · Alexander Wolff

Lehrstuhl f¨ ur Informatik I Universit¨ at W¨ urzburg, Germany

ESA’11

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Minimum Manhattan Networks

Given a set of points called terminals in Rd, find a minimum-length network such that each pair of terminals is connected by a Manhattan path. terminals

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Minimum Manhattan Networks

Given a set of points called terminals in Rd, find a minimum-length network such that each pair of terminals is connected by a Manhattan path. terminals minimum Manhattan network

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Minimum Manhattan Networks

Given a set of points called terminals in Rd, find a minimum-length network such that each pair of terminals is connected by a Manhattan path. terminals minimum Manhattan network

A Manhattan path is a chain of axis-parallel line segments whose total length is the Manhattan distance of the chain’s endpoints.

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SLIDE 5

Previous Results

Results for 2D

  • introduced by Gudmundsson et al. (NJC’01)
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SLIDE 6

Previous Results

Results for 2D

  • introduced by Gudmundsson et al. (NJC’01)
  • currently best approximation ratio is 2;

by Nouiua (’05), Chepoi et al. (’08), Guo et al. (’08) – using different techniques

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SLIDE 7

Previous Results

Results for 2D

  • introduced by Gudmundsson et al. (NJC’01)
  • currently best approximation ratio is 2;

by Nouiua (’05), Chepoi et al. (’08), Guo et al. (’08) – using different techniques

  • NP-hardness shown by Chin et al. (SoCG’09)
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SLIDE 8

Previous Results

Results for 2D

  • introduced by Gudmundsson et al. (NJC’01)
  • currently best approximation ratio is 2;

by Nouiua (’05), Chepoi et al. (’08), Guo et al. (’08) – using different techniques

  • NP-hardness shown by Chin et al. (SoCG’09)

Results for 3D (or higher dimensions)

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SLIDE 9

Previous Results

Results for 2D

  • introduced by Gudmundsson et al. (NJC’01)
  • currently best approximation ratio is 2;

by Nouiua (’05), Chepoi et al. (’08), Guo et al. (’08) – using different techniques

  • NP-hardness shown by Chin et al. (SoCG’09)

Results for 3D (or higher dimensions)

  • constant factor approximation for very restricted 3D case

by Mu˜ noz et al. (WALCOM’09)

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SLIDE 10

Previous Results

Results for 2D

  • introduced by Gudmundsson et al. (NJC’01)
  • currently best approximation ratio is 2;

by Nouiua (’05), Chepoi et al. (’08), Guo et al. (’08) – using different techniques

  • NP-hardness shown by Chin et al. (SoCG’09)

Results for 3D (or higher dimensions)

  • constant factor approximation for very restricted 3D case

by Mu˜ noz et al. (WALCOM’09)

  • Non-trivial approximations for unrestricted version?
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SLIDE 11

Our Results

  • 4(k − 1) approximation for 3D –

if the terminals lie in the union of k horizontal planes

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SLIDE 12

Our Results

  • 4(k − 1) approximation for 3D –

if the terminals lie in the union of k horizontal planes

  • O(nǫ) approximation for general case

in any fixed dimension and for any fixed ǫ > 0

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SLIDE 13

Our Results

  • 4(k − 1) approximation for 3D –

if the terminals lie in the union of k horizontal planes

  • O(nǫ) approximation for general case

in any fixed dimension and for any fixed ǫ > 0

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SLIDE 14

Decomposition into Directional Subproblems

Directional Subproblem: M-connect all pairs of terminals t = (x, y, z) and t′ = (x′, y ′, z′) with x ≤ x′, y ≤ y ′, z ≤ z′ . t t′

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Decomposition into Directional Subproblems

Directional Subproblem: M-connect all pairs of terminals t = (x, y, z) and t′ = (x′, y ′, z′) with x ≤ x′, y ≤ y ′, z ≤ z′ . t t′ We call such pairs relevant.

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SLIDE 16

Decomposition into Directional Subproblems

Directional Subproblem: M-connect all pairs of terminals t = (x, y, z) and t′ = (x′, y ′, z′) with x ≤ x′, y ≤ y ′, z ≤ z′ . t t′ General problem can be decomposed into four directional subproblems We call such pairs relevant.

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Two Horizontal Planes

B R Let N be some directional Manhattan network.

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Two Horizontal Planes

B R Let N be some directional Manhattan network. horizontal part Nxy

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SLIDE 19

Two Horizontal Planes

B R Let N be some directional Manhattan network. horizontal part Nxy vertical part Nz ”pillars”

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Two Horizontal Planes

B R Let N be some directional Manhattan network. horizontal part Nxy vertical part Nz ”pillars” project onto x–y plane

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SLIDE 21

2D Projection

pillar ∈ Nz Nxy Legend

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SLIDE 22

2D Projection

pillar ∈ Nz Nxy

  • n top plane
  • n bottom plane

Legend

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SLIDE 23

2D Projection

pillar ∈ Nz Nxy

  • n top plane
  • n bottom plane

Nxy is a directional 2D Manhattan network for R ∪ B Legend

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SLIDE 24

2D Projection

pillar ∈ Nz Nxy

  • n top plane
  • n bottom plane

Use 2D approximation

  • n both planes

Nxy is a directional 2D Manhattan network for R ∪ B Legend

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Approximating the Horizontal Part is Easy

B R Copy 2-approximate 2D network for R ∪ B onto both planes

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SLIDE 26

But How to Find the Pillars?

pillar ∈ Nz Nxy Each rectangle spanned by a relevant red-blue terminal pair is pierced by some pillar in N.

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But How to Find the Pillars?

pillar ∈ Nz Nxy Each rectangle spanned by a relevant red-blue terminal pair is pierced by some pillar in N.

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SLIDE 28

Lower Bounding by Red-Blue Piercings

Given a set of red and blue points in the plane, find a minimum set of piercing pts (pillars) such that each rectangle spanned by a relevant red-blue pair is pierced. Subproblem RBP:

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Lower Bounding by Red-Blue Piercings

Given a set of red and blue points in the plane, find a minimum set of piercing pts (pillars) such that each rectangle spanned by a relevant red-blue pair is pierced. OPTRBP ≤ #pillars in Nz. Subproblem RBP:

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SLIDE 30

Lower Bounding by Red-Blue Piercings

Given a set of red and blue points in the plane, find a minimum set of piercing pts (pillars) such that each rectangle spanned by a relevant red-blue pair is pierced. OPTRBP ≤ #pillars in Nz. Subproblem RBP: Theorem (Soto & Telha, IPCO’11) Red-blue piercing can be solved in polynomial time.

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SLIDE 31

Converting Piercings to Pillars (I)

Given red-blue piercing S and Manhattan network for R ∪ B, we can move the needles (pts) in S so that for each relevant pair (r, b) there is an M-path that contains a needle of S. Lemma

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Converting Piercings to Pillars (I)

Given red-blue piercing S and Manhattan network for R ∪ B, we can move the needles (pts) in S so that for each relevant pair (r, b) there is an M-path that contains a needle of S. Lemma

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SLIDE 33

Converting Piercings to Pillars (I)

Given red-blue piercing S and Manhattan network for R ∪ B, we can move the needles (pts) in S so that for each relevant pair (r, b) there is an M-path that contains a needle of S. Lemma

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SLIDE 34

Converting Piercings to Pillars (I)

Given red-blue piercing S and Manhattan network for R ∪ B, we can move the needles (pts) in S so that for each relevant pair (r, b) there is an M-path that contains a needle of S. Lemma

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SLIDE 35

Converting Piercings to Pillars (II)

B R

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SLIDE 36

Converting Piercings to Pillars (II)

B R Extend piercing pts to pillars

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SLIDE 37

Converting Piercings to Pillars (II)

B R Extend piercing pts to pillars

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SLIDE 38

Converting Piercings to Pillars (II)

B R Extend piercing pts to pillars feasible 3D Manhattan network!

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SLIDE 39

Converting Piercings to Pillars (II)

B R Extend piercing pts to pillars feasible 3D Manhattan network! cost ≤ 4 · OPT (due to the four directions)

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k Planes – Horizontal Part

copy 2D Manhattan network onto each plane

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SLIDE 41

k Planes – Horizontal Part

copy 2D Manhattan network onto each plane

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k Planes – Vertical Part

i i + 1 k 1 . . . . . . Bi Ri

 

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SLIDE 43

k Planes – Vertical Part

i i + 1 k 1 . . . . . . Bi Ri

  • Choose i such that (Ri, Bi) can be pierced with a minimum number of pillars.

 

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SLIDE 44

k Planes – Vertical Part

i i + 1 k 1 . . . . . . Bi Ri

  • Choose i such that (Ri, Bi) can be pierced with a minimum number of pillars.
  • Extend those pillars over all k planes.

 

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SLIDE 45

k Planes – Vertical Part

i i + 1 k 1 . . . . . . Bi Ri

  • Choose i such that (Ri, Bi) can be pierced with a minimum number of pillars.
  • Extend those pillars over all k planes.

 

⇒ cost ≤ OPTz.

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SLIDE 46

k Planes – Vertical Part

i i + 1 k 1 . . . . . . Bi Ri

  • Choose i such that (Ri, Bi) can be pierced with a minimum number of pillars.
  • Extend those pillars over all k planes.
  • All terminal pairs r ∈ Ri, b ∈ Bi are M-connected by v-part ∪ h-part

 

⇒ cost ≤ OPTz.

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SLIDE 47

k Planes – Vertical Part

i i + 1 k 1 . . . . . . Bi Ri

  • Choose i such that (Ri, Bi) can be pierced with a minimum number of pillars.
  • Extend those pillars over all k planes.
  • All terminal pairs r ∈ Ri, b ∈ Bi are M-connected by v-part ∪ h-part

 

  • ⇒ cost ≤ OPTz.
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SLIDE 48

k Planes – Vertical Part

i i + 1 k 1 . . . . . . Bi Ri

  • Choose i such that (Ri, Bi) can be pierced with a minimum number of pillars.
  • Extend those pillars over all k planes.
  • All terminal pairs r ∈ Ri, b ∈ Bi are M-connected by v-part ∪ h-part
  • Apply this recursively to planes (1, . . . , i) and (i + 1, . . . , k).

 

  • ⇒ cost ≤ OPTz.
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SLIDE 49

k Planes – Vertical Part

i i + 1 k 1 . . . . . . Bi Ri

  • Choose i such that (Ri, Bi) can be pierced with a minimum number of pillars.
  • Extend those pillars over all k planes.
  • All terminal pairs r ∈ Ri, b ∈ Bi are M-connected by v-part ∪ h-part
  • Apply this recursively to planes (1, . . . , i) and (i + 1, . . . , k).

 

  • ⇒ cost ≤ OPTz.
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SLIDE 50

k Planes – Vertical Part

i i + 1 k 1 . . . . . . Bi Ri

  • Choose i such that (Ri, Bi) can be pierced with a minimum number of pillars.
  • Extend those pillars over all k planes.
  • All terminal pairs r ∈ Ri, b ∈ Bi are M-connected by v-part ∪ h-part
  • Apply this recursively to planes (1, . . . , i) and (i + 1, . . . , k).
  • Ratio satisfies r(k) ≤ r(i) + r(k − i − 1) + 1.

 

  • ⇒ cost ≤ OPTz.
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SLIDE 51

k Planes – Vertical Part

i i + 1 k 1 . . . . . . Bi Ri

  • Choose i such that (Ri, Bi) can be pierced with a minimum number of pillars.
  • Extend those pillars over all k planes.
  • All terminal pairs r ∈ Ri, b ∈ Bi are M-connected by v-part ∪ h-part
  • Apply this recursively to planes (1, . . . , i) and (i + 1, . . . , k).
  • Ratio satisfies r(k) ≤ r(i) + r(k − i − 1) + 1.

 

  • ⇒ r(k) ≤ k − 1.

⇒ cost ≤ OPTz.

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SLIDE 52

k Planes – Vertical Part

i i + 1 k 1 . . . . . . Bi Ri

  • Choose i such that (Ri, Bi) can be pierced with a minimum number of pillars.
  • Extend those pillars over all k planes.
  • All terminal pairs r ∈ Ri, b ∈ Bi are M-connected by v-part ∪ h-part
  • Apply this recursively to planes (1, . . . , i) and (i + 1, . . . , k).
  • Ratio satisfies r(k) ≤ r(i) + r(k − i − 1) + 1.

Overall ratio 4(k − 1)

 

  • ⇒ r(k) ≤ k − 1.

⇒ cost ≤ OPTz.

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SLIDE 53

k Planes – Vertical Part

i i + 1 k 1 . . . . . . Bi Ri

  • Choose i such that (Ri, Bi) can be pierced with a minimum number of pillars.
  • Extend those pillars over all k planes.
  • All terminal pairs r ∈ Ri, b ∈ Bi are M-connected by v-part ∪ h-part
  • Apply this recursively to planes (1, . . . , i) and (i + 1, . . . , k).
  • Ratio satisfies r(k) ≤ r(i) + r(k − i − 1) + 1.

Overall ratio 4(k − 1)

 

  • ⇒ r(k) ≤ k − 1.

⇒ cost ≤ OPTz.

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SLIDE 54

Our Results for Higher Dimensions

  • 4(k − 1) approximation for 3D –

if the terminals lie in the union of k horizontal planes

  • O(nǫ) approximation for general case

in any fixed dimension and for any fixed ǫ > 0

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SLIDE 55

Grid Algorithm for General Case

  • determine bounding cuboid
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Grid Algorithm for General Case

  • determine bounding cuboid
  • partition into c × c slabs with n/c terminals each
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SLIDE 57

Grid Algorithm for General Case

  • determine bounding cuboid
  • partition into c × c slabs with n/c terminals each
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SLIDE 58

Grid Algorithm for General Case

  • determine bounding cuboid
  • partition into c × c slabs with n/c terminals each
  • add resulting grid to solution
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SLIDE 59

Grid Algorithm for General Case

  • determine bounding cuboid
  • partition into c × c slabs with n/c terminals each
  • add resulting grid to solution
  • connect terminals to grid
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SLIDE 60

Grid Algorithm for General Case

  • determine bounding cuboid
  • partition into c × c slabs with n/c terminals each
  • add resulting grid to solution
  • connect terminals to grid
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SLIDE 61

Grid Algorithm for General Case

  • determine bounding cuboid
  • partition into c × c slabs with n/c terminals each
  • add resulting grid to solution
  • connect terminals to grid
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SLIDE 62

Grid Algorithm for General Case

  • determine bounding cuboid
  • partition into c × c slabs with n/c terminals each
  • add resulting grid to solution
  • connect terminals to grid

(”patching”by directed Steiner trees)

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SLIDE 63

Grid Algorithm for General Case

  • determine bounding cuboid
  • partition into c × c slabs with n/c terminals each
  • add resulting grid to solution
  • connect terminals to grid

(”patching”by directed Steiner trees)

  • pairs in different slabs are M-connected
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SLIDE 64

Grid Algorithm for General Case

  • determine bounding cuboid
  • partition into c × c slabs with n/c terminals each
  • add resulting grid to solution
  • connect terminals to grid

(”patching”by directed Steiner trees)

  • pairs in different slabs are M-connected
  • apply recursively to slabs
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SLIDE 65

Grid Algorithm for General Case

  • determine bounding cuboid
  • partition into c × c slabs with n/c terminals each
  • add resulting grid to solution
  • connect terminals to grid

(”patching”by directed Steiner trees)

  • pairs in different slabs are M-connected
  • apply recursively to slabs
  • overall ratio O(nǫ)

(by choosing c accordingly)

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SLIDE 66

Open Questions

  • Can we achieve O(log n) or even constant ratio?
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SLIDE 67

Open Questions

  • Can we achieve O(log n) or even constant ratio?
  • “GMMN”:

What if only a given set of terminal pairs needs to be M-connected? (Open question of Chepoi; unknown even for 2D.)

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SLIDE 68

Open Questions

  • Can we achieve O(log n) or even constant ratio?
  • “GMMN”:

What if only a given set of terminal pairs needs to be M-connected? (Open question of Chepoi; unknown even for 2D.)

Latest News

  • O(logd+1 n)-approximation algorithm for d dimensions.
  • O(log n)-approximation algorithm for 2D.
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SLIDE 69

Open Questions

  • Can we achieve O(log n) or even constant ratio?
  • “GMMN”:

What if only a given set of terminal pairs needs to be M-connected? (Open question of Chepoi; unknown even for 2D.)

Latest News

  • O(logd+1 n)-approximation algorithm for d dimensions.
  • O(log n)-approximation algorithm for 2D.
  • Both these results hold for GMMN as well.