New Network Design with Coverage Costspages.cs.wisc.edu/~seeun/presentations/approx14.pdf · 2014. 10. 16. · Network design with economies of scale. å edges Cost of edge Load on

Network Design withCoverage Costs

Siddharth Barman 1 Shuchi Chawla 2 Seeun William Umboh 2

1Caltech

2University of Wisconsin-Madison

APPROX-RANDOM 2014

Motivation

Physical Flow vs Data Flow

vs.

Commodity Network Internet

Unlike physical commodities, data can be easily duplicated,compressed, and combined.

Physical Flow vs Data Flow

Flow = 1

Flow = 2

P = 10010

P = 10010vs.

Unlike physical commodities, data can be easily duplicated,compressed, and combined.

Why Coverage Costs?

• Want a cost structure that captures bandwidth savings obtainedby eliminating redundancy in data.

▶ Packet deduplication capability exists in networks today.

Problem Statement

Load Function

We focus on redundant-data elimination:

Load on an edge ℓe = # distinct data packets on it.

P1 : 11100P1 : 11100P2 : 00011

Load = 2

Input:• Graph with edge costs, g terminal pairs (s1, t1), . . . ,(sg, tg)

• A global set of packets Π.• A demand set Dj ⊂ Π for each j.

Goal: Find (sj, tj) path on which to route Dj for each j minimizing

s1 t1

s2 t2

Input:• Graph with edge costs, g terminal pairs (s1, t1), . . . ,(sg, tg)• A global set of packets Π.

• A demand set Dj ⊂ Π for each j.

�s1 t1

s2 t2

Input:• Graph with edge costs, g terminal pairs (s1, t1), . . . ,(sg, tg)• A global set of packets Π.• A demand set Dj ⊂ Π for each j.

�

D1D2

s1 t1

s2 t2

∑edges

Cost of edge×Load on edge

�

D1D2

s1 t1

s2 t2

∑edges

Cost of edge×# distinct packets on edge

�

D1D2

s1 t1

s2 t2

∑edges

�

D1D2

s1 t1

s2 t2

∑edges

�

D1D2

s1 t1

s2 t2

|D1|

|D2|

|D1|

|D2||D1 U D2|

More generally, we consider terminal groups.Input:

• Graph with edge costs, g terminal groups X1, . . . ,Xg ⊂ V.• A global set of packets Π.• A demand set Dj ⊂ Π for each j.

Goal: Find a Steiner tree for Xj on which to broadcast Dj for each jminimizing

∑edges

Special cases (for terminal pairs):

• Pairwise disjoint demands =⇒ shortest-paths

s1 t1

s2 t2

|D1|

|D2|

�

D1 D2

• Identical demands =⇒ Steiner forests1 t1

s2 t2

|D|

|D||D|

�

D1, D2 = D

Special cases (for terminal pairs):

• Pairwise disjoint demands =⇒ shortest-paths

s1 t1

s2 t2

|D1|

|D2|

�

D1 D2

• Identical demands =⇒ Steiner forests1 t1

s2 t2

|D|

|D||D|

�

D1, D2 = D

Challenges

• Intuitively, difficulty depends on complexity of demand sets

�

• Desire an approximation in terms of g (number of groups), e.g.O(logg) (tree embeddings only yield O(logn)).

Challenges

• Intuitively, difficulty depends on complexity of demand sets

�

• Desire an approximation in terms of g (number of groups), e.g.O(logg) (tree embeddings only yield O(logn)).

Our results

Laminar DemandsThe family of demand sets is said to be laminar if ∀ i, j one of thefollowing holds: Di ∩Dj = ϕ or Di ⊂ Dj or Dj ⊂ Di.

⇧

D1 D2

D3

D4D5

⇧

D2D1

D3 D5D4

TheoremNDCC with laminar demands admits a 2-approximation.

Previous work: O(log |Π|) [Barman-Chawla ’12].

Laminar DemandsThe family of demand sets is said to be laminar if ∀ i, j one of thefollowing holds: Di ∩Dj = ϕ or Di ⊂ Dj or Dj ⊂ Di.

⇧

D1 D2

D3

D4D5

⇧

D2D1

D3 D5D4

TheoremNDCC with laminar demands admits a 2-approximation.

Previous work: O(log |Π|) [Barman-Chawla ’12].

Sunflower DemandsThe family of demand sets is said to be a sunflower family if thereexists a core C ⊂ Π such that ∀ i, j, Di ∩Dj = C.

C

Di \ C

Dj \ C

No O(log1/4−γ g)-approximation for any γ > 0 (reduction fromsingle-cable buy-at-bulk [Andrews-Zhang ’02]).

TheoremNDCC with sunflower demands admits an O(log g) approximation overunweighted graphs with V =

∪j Xj.

C

Di \ C

Dj \ C

∪j Xj.

C

Di \ C

Dj \ C

∪j Xj.

Related WorkNetwork design with economies of scale.

∑edges

• Submodular costs on edges [Hayrapetyan-Swamy-Tardos ’05]▶ O(logn) via tree embeddings.

• Buy-at-Bulk Network Design [Salman et al. ’97]▶ Single-source: O(1) [Guha et al. ’01, Talwar ’02,

Gupta-Kumar-Roughgarden ’03]▶ Multiple-source: Ω(log

14 n) hardness [Andrews ’04]

∑edges

Approach: Laminar

Laminar⇧

D1 D2

D3

D4D5

⇧

D2D1

D3 D5D4

• Naive LP does not have much structure• Key idea: exploit laminarity to write a different LP• Run an extension of AKR-GW primal-dual algorithm to get

2-approx

Approach: Sunflower(single-source for this talk)

Sunflower Family of DemandsThe family of demand sets is said to be a sunflower family if thereexists C ⊂ Π such that ∀ i, j, Di ∩Dj = C.

C

Di \ C

Dj \ C

TheoremNetwork Design with Coverage Costs in the sunflower demands settingadmits an O(log g) approximation over unweighted graphs with V =

∪j Xj.

Structure of OPTDefine:

• E∗j = Steiner tree for Xj on which OPT broadcasts Dj ⊃ C• E∗0 =

∪j E

∗j

Note: E∗0 spans V because V =∪

j Xj and single-source (each Xjcontains s)

c(OPT) = |C| · c(E∗0)+∑j|Dj \C| · c(E∗j )

≥ |C| · c(MST)+∑j|Dj \C| · c(opt Steiner for Xj)

∪j E

∗j

c(OPT) = |C| · c(E∗0)+∑j|Dj \C| · c(E∗j )

∪j E

∗j

c(OPT) = |C| · c(E∗0)+∑j|Dj \C| · c(E∗j )

Group Spanners

DefinitionFor a graph G and g groups X1, . . . ,Xg ⊂ V, a subgraph H is a (α,β )group spanner if

• c(H)≤ αc(MST),• c(opt Steiner for Xj in H)≤ β c(opt Steiner for Xj in G) for all j.

Usual spanners: Xjs are all vertex pairs.

• c(shortest u− v path in H)≤ β c(shortest u− v path in G) for allu,v ∈ V.

Group Spanners

DefinitionFor a graph G and g groups X1, . . . ,Xg ⊂ V, a subgraph H is a (α,β )group spanner if

• c(H)≤ αc(MST),• c(opt Steiner for Xj in H)≤ β c(opt Steiner for Xj in G) for all j.

Usual spanners: Xjs are all vertex pairs.

• c(shortest u− v path in H)≤ β c(shortest u− v path in G) for allu,v ∈ V.

Using Group Spanners

Given (α,β ) group spanner H, route Dj along 2-approximate Steinertree for Xj in H.

Cost ≤ |C| · c(H)+∑j|Dj \C| ·2c(opt Steiner for Xj in H)

≤ |C| ·αc(MST)+∑j|Dj \C| ·2β c(opt Steiner for Xj)

≤ max{α,2β}OPT .

Group Spanners Result

LemmaGiven an unweighted graph G and g groups X1, . . . ,Xg ⊂ V with V =

∪j Xj,

we can construct in polynomial time a (O(1),O(log g)) group spanner.

TheoremNetwork Design with Coverage Costs in the sunflower demands settingadmits an O(log g) approximation over unweighted graphs with V =

∪j Xj.

Summary• Coverage cost model for designing information networks.

P1 : 11100P1 : 11100P2 : 00011

Load = 2

• Laminar demands: 2-approximation via primal-dual.⇧

D1 D2

D3

D4D5

⇧

D2D1

D3 D5D4

• Sunflower demands: O(log g)-approximation (unweighted graph,no Steiner vertices).

C

Di \ C

Dj \ C

• Group spanners: (O(1),O(log g)) group spanners (unweightedgraph, no Steiner vertices).

P1 : 11100P1 : 11100P2 : 00011

Load = 2

D1 D2

D3

D4D5

⇧

D2D1

D3 D5D4

C

Di \ C

Dj \ C

P1 : 11100P1 : 11100P2 : 00011

Load = 2

D1 D2

D3

D4D5

⇧

D2D1

D3 D5D4

C

Di \ C

Dj \ C

Open Problems

• Group spanners: (O(log g),O(log g)) in general?

• Sunflower demands: O(log g) approximation in general?• Terminal pairs with single-source: O(1)?

Thanks!

Open Problems

• Group spanners: (O(log g),O(log g)) in general?• Sunflower demands: O(log g) approximation in general?

• Terminal pairs with single-source: O(1)?

Thanks!

Open Problems

• Group spanners: (O(log g),O(log g)) in general?• Sunflower demands: O(log g) approximation in general?• Terminal pairs with single-source: O(1)?

Thanks!

Open Problems

• Group spanners: (O(log g),O(log g)) in general?• Sunflower demands: O(log g) approximation in general?• Terminal pairs with single-source: O(1)?

Thanks!

Documents

New Network Design with Coverage Costspages.cs.wisc.edu/~seeun/presentations/approx14.pdf · 2014. 10. 16. · Network design with economies of scale. å edges Cost of edge Load on