Network Design with Coverage Costs

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Network Design with

Coverage Costs

Siddharth Barman1 Shuchi Chawla2 Seeun William Umboh2

1_Caltech

2_{University of Wisconsin-Madison}

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Physical Flow vs Data Flow

vs.

Commodity Network Internet

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

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Physical Flow vs Data Flow

Flow = 1 Flow = 1 Flow = 2 P= 10010 P= 10010 P= 10010

vs.

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

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Why Coverage Costs?

• Want a cost structure that captures bandwidth savings obtained by eliminatingredundancy in data.

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Load Function

We focus on redundant-data elimination:

Load on an edgeℓe =#distinctdata packets on it.

P1: 11100

P2: 00011

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Input:

• Graph with edge costs,gterminal pairs(s1,t1), . . . ,(sg,tg)

• A global set of packetsΠ.

• A demand setDj⊂Πfor eachj.

Goal: Find(sj,tj)path on which to routeDjfor eachjminimizing

s1 t1

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Input:

s1 t1

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Input:

D1 D2

s1 t1

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Input:

∑

edges

Cost of edge×Load on edge

D1 D2

s1 t1

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Input:

∑

edges

Cost of edge×#distinctpackets on edge

D1 D2

s1 t1

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Input:

∑

edges

D1 D2

s1 t1

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Input:

∑

edges

D1 D2 s1 t1 s2 t2 |D1| |D2| |D1| |D2| |D1 U D2|

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More generally, we consider terminalgroups.

Input:

• Graph with edge costs,gterminal groupsX1, . . . ,Xg⊂V.

Goal: Find a Steiner tree forXj on which to broadcastDjfor eachj

minimizing

∑

edges

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Special cases (for terminal pairs):

• Pairwise disjoint demands =⇒ shortest-paths

s1 t1

s2 t2

|D1|

|D2| D1 D2

• Identical demands =⇒ Steiner forest

s1 t1 s2 t2 |D| |D| |D| |D| |D| D1, D2 = D

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Special cases (for terminal pairs):

• Pairwise disjoint demands =⇒ shortest-paths

s1 t1

s2 t2

|D1|

|D2| D1 D2

• Identical demands =⇒ Steiner forest

s1 t1 s2 t2 |D| |D| |D| |D| |D| D1, D2 = D

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Challenges

• Intuitively, difﬁculty depends on complexity of demand sets

• Desire an approximation in terms ofg(number of groups), e.g.

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Challenges

• Intuitively, difﬁculty depends on complexity of demand sets

• Desire an approximation in terms ofg(number of groups), e.g.

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Laminar Demands

The family of demand sets is said to belaminarif∀i,jone of the following holds: Di∩Dj=ϕor Di⊂Dj orDj⊂Di.

⇧

D1 D2 D3 D4 D5 ⇧ D2 D1 D3 D5 D4 Theorem

NDCC with laminar demands admits a2-approximation.

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Laminar Demands

The family of demand sets is said to belaminarif∀i,jone of the following holds: Di∩Dj=ϕor Di⊂Dj orDj⊂Di.

⇧

D1 D2 D3 D4 D5 ⇧ D2 D1 D3 D5 D4 Theorem

NDCC with laminar demands admits a2-approximation.

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Sunﬂower Demands

The family of demand sets is said to be asunﬂowerfamily if there exists acoreC⊂Πsuch that∀i,j,Di∩Dj=C.

C

Di \ C Dj \ C

NoO(log1/4−γg)-approximation for anyγ>0 (reduction from single-cable buy-at-bulk[Andrews-Zhang ’02]).

Theorem

NDCC with sunﬂower demands admits an O(log g)approximation over

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Sunﬂower Demands

C

Di \ C Dj \ C

Theorem

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Sunﬂower Demands

C

Di \ C Dj \ C

Theorem

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Related Work

Network 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]

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Related Work

∑

edges

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Related Work

∑

edges

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Laminar

⇧

D1 D2 D3 D4 D5 ⇧ D2 D1 D3 D5 D4

• 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

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Approach: Sunﬂower

(single-source for this talk)

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Sunﬂower Family of Demands

The family of demand sets is said to be asunﬂowerfamily if there existsC⊂Πsuch that∀i,j,Di∩Dj=C.

C

Di \ C Dj \ C

Theorem

Network Design with Coverage Costs in the sunﬂower demands setting

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Structure of OPT

Deﬁne:

• E∗_j =Steiner tree forXj on which OPT broadcastsDj⊃C

• E∗₀=∪_jE∗_j

Note:E∗₀spansVbecauseV=∪_jXjand single-source (eachXj

containss) c(OPT) =|C|·c(E∗₀) +

∑

j |Dj\C|·c(E∗j) ≥|C|·c(MST) +

∑

j

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Structure of OPT

Deﬁne:

• E∗₀=∪_jE∗_j

∑

j |Dj\C|·c(E∗j) ≥|C|·c(MST) +

∑

j

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Structure of OPT

Deﬁne:

• E∗₀=∪_jE∗_j

∑

j |Dj\C|·c(E∗j) ≥|C|·c(MST) +

∑

j

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Group Spanners

Deﬁnition

For a graphGandggroupsX1, . . . ,Xg⊂V, a subgraphHis a(α,β)

group spannerif

• c(H)≤αc(MST),

• c(opt Steiner forXj inH)≤βc(opt Steiner forXj inG)for allj.

Usual spanners: Xjs are all vertex pairs.

• c(shortestu−vpath inH)≤βc(shortestu−vpath inG)for all

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Group Spanners

Deﬁnition

For a graphGandggroupsX1, . . . ,Xg⊂V, a subgraphHis a(α,β)

group spannerif

• c(H)≤αc(MST),

• c(opt Steiner forXj inH)≤βc(opt Steiner forXj inG)for allj.

Usual spanners: Xjs are all vertex pairs.

• c(shortestu−vpath inH)≤βc(shortestu−vpath inG)for all

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Using Group Spanners

Given(α,β)group spannerH, routeDj along 2-approximate Steiner

tree forXj inH.

Cost≤ |C| ·c(H) +

∑

j

|Dj\C| ·2c(opt Steiner forXjinH) ≤ |C| ·αc(MST) +

∑

j

|Dj\C| ·2βc(opt Steiner forXj) ≤max{α,2β}OPT.

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Group Spanners Result

Lemma

Given anunweightedgraph G and g groups X1, . . . ,Xg⊂V withV=

∪

jXj,

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

Theorem

Network Design with Coverage Costs in the sunﬂower demands setting

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Summary

• Coverage cost modelfor designing information networks.

P1: 11100 P1: 11100

P2: 00011

Load = 2

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

⇧ D1 D2 D3 D4 D5 ⇧ D2 D1 D3 D5 D4

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

C Di \ C

Dj \ C

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

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Summary

P1: 11100 P1: 11100

P2: 00011

Load = 2

⇧ D1 D2 D3 D4 D5 ⇧ D2 D1 D3 D5 D4

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

C Di \ C

Dj \ C

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Summary

P1: 11100 P1: 11100

P2: 00011

Load = 2

⇧ D1 D2 D3 D4 D5 ⇧ D2 D1 D3 D5 D4

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

C Di \ C

Dj \ C

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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)?

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Open Problems

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Open Problems

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