Approximating the Traffic Grooming Problem

Mordo ShalomTel Hai Academic College & Technion

Approximating the Traffic Grooming Problem 2

Joint work with

Michele Flammini – L’AquilaLuca Moscardelli – L’AquilaShmuel Zaks - Technion

Approximating the Traffic Grooming Problem 3

Outline

Optical networks The Min ADM Problem The Traffic Grooming

Problem Algorithm GROOMBYSC

Approximating the Traffic Grooming Problem 4

Outline

Optical networks The Min ADM Problem The Traffic Grooming

Problem Algorithm GROOMBYSC

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The MIN ADM Problem

W=2, ADM=4 W=1, ADM=3

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W-ADM tradeoff

W=2, ADM=8 W=3, ADM=7

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The Goal

Given a set of lightpaths, find a valid coloring with minimum number of ADMs.

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Outline

Optical networks The Min ADM Problem The Traffic Grooming

Problem Algorithm GROOMBYSC

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The Traffic Grooming Problem

A generalization of the MIN ADM problem.

Instead of requests for entire lightpaths, the input contains requests for integer multiples of 1/g of one lighpath’s bandwidth.

g is an integer given with the instance.

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The Traffic Grooming Problem

W=2, ADM=8 W=1, ADM=7

g=2

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The Goal

Given a set of requests and a grooming factor g, find a valid coloring with minimum number of ADMs.

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Notation & Immediate Results P: The set of paths. SOL: The # of ADMs used by a

solution. OPT: The # of ADMs used by an

optimal solution.|P|/g SOL 2|P||P|/g OPT 2|P|rSOL = SOL/OPT 2g

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Outline

Optical networks The Min ADM Problem The Traffic Grooming

Problem Algorithm GROOMBYSC

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Main Resultg > 1, Ring Networks:

General traffic:

An O(log g) approximation algorithm for any fixed g.

Can be used in general networks

Analysis can be extended to some other topologies.

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Approximation algorithm (log g)

¬ ÈS S {A}

Input: Graph G, set of lightpaths P, g > 0

Step 1: Choose a parameter k = k(g).

Step 2: Consider all subsets of P of size

If a subset A is 1-colorable (i.e., any edge is used at most g times) then

weight[A]=endpoints(A);

£ ×k g

¬ ÆS

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Algorithm (cont’d)

Step 3: COVER(an approximation to) the Minimum Weight Set Cover of S[], weight[], using [Chvatal79]

Step 4: Convert COVER to a PARTITION

PARTITION induces a coloring of the paths

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Analysis

Let , then:

If B is 1-colorable then A is 1-colorable (correctness).

Cost(A) Cost(B).

A B

Therefore: …

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k g

cost(PARTI TI ON)

weight(COVER)

H weight(MI NCOVER)

(1+ln(k g))w

ALG=

Sh Ceig t( )

for every set cover SC.

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Lemma: There is a set cover SC, s.t.: 2g

weight( ) 1+SC Pk

O T

(1+ln(k g)) weightA LG (S C)

for any set cover SC.

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k g

weight(COVER)

H weight(MI NCOVER)

(1+ln(k g))weight( )

2g(1+ln(k g)

A

) 1+

SC

k

LG

OPT

Conclusion:

For k = g ln g : 2lng +o(lngA G )L O PT

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Proof of Lemma

Lemma: There is a set cover SC, s.t.: 2g

weight( ) 1+SC Pk

O T

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Proof of LemmaConsider a color l of OPT.Consider the set Pl of paths

colored l.Consider the set of ADMs

operating at wavelength l. (i.e. endpoints(Pl) )

Divide endpoints(Pl) into sets of k consecutive nodes.

For simplicity assume |endpoints(Pl)|=m.k

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k k k k

1weight[S ] k +gS1 S2 Sm

M=4 k=6

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Analysis (cont’d),

,1

,1

[ ]

[ ] ( )

.

[ ] 1

i

m

ii

m

ii

weight S k g

weight S m k g

OPT m k

gweight S OPT

k

w/o the assumption we have:

,1

2[ ] 1

m

ii

gweight S OPT

k

,1

2[ ] 1

m

ii

gweight S OPT

k

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Analysis (cont’d)

,iS S

,iS S

,, | | .ii S k g

and also 1-colorable thus

,,

ii

P S

Moreover

,iSC S Therefore

Is a set cover with sets from S.

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