Approximating the Traffic Grooming Problem
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Approximating the Traffic Grooming Problem
Mordo ShalomTel Hai Academic College & Technion
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Approximating the Traffic Grooming Problem 2
Joint work with
Michele Flammini – L’AquilaLuca Moscardelli – L’AquilaShmuel Zaks - Technion
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Approximating the Traffic Grooming Problem 3
Outline
Optical networks The Min ADM Problem The Traffic Grooming
Problem Algorithm GROOMBYSC
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Approximating the Traffic Grooming Problem 4
Outline
Optical networks The Min ADM Problem The Traffic Grooming
Problem Algorithm GROOMBYSC
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Approximating the Traffic Grooming Problem 5
The MIN ADM Problem
W=2, ADM=4 W=1, ADM=3
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Approximating the Traffic Grooming Problem 6
W-ADM tradeoff
W=2, ADM=8 W=3, ADM=7
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Approximating the Traffic Grooming Problem 7
The Goal
Given a set of lightpaths, find a valid coloring with minimum number of ADMs.
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Approximating the Traffic Grooming Problem 8
Outline
Optical networks The Min ADM Problem The Traffic Grooming
Problem Algorithm GROOMBYSC
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Approximating the Traffic Grooming Problem 9
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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Approximating the Traffic Grooming Problem 10
The Traffic Grooming Problem
W=2, ADM=8 W=1, ADM=7
g=2
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Approximating the Traffic Grooming Problem 11
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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Approximating the Traffic Grooming Problem 12
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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Approximating the Traffic Grooming Problem 13
Outline
Optical networks The Min ADM Problem The Traffic Grooming
Problem Algorithm GROOMBYSC
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Approximating the Traffic Grooming Problem 14
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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Approximating the Traffic Grooming Problem 15
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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Approximating the Traffic Grooming Problem 16
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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Approximating the Traffic Grooming Problem 17
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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Approximating the Traffic Grooming Problem 18
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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Approximating the Traffic Grooming Problem 19
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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Approximating the Traffic Grooming Problem 20
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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Approximating the Traffic Grooming Problem 21
Proof of Lemma
Lemma: There is a set cover SC, s.t.: 2g
weight( ) 1+SC Pk
O T
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Approximating the Traffic Grooming Problem 22
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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Approximating the Traffic Grooming Problem 23
k k k k
1weight[S ] k +gS1 S2 Sm
M=4 k=6
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Approximating the Traffic Grooming Problem 24
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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Approximating the Traffic Grooming Problem 25
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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Approximating the Traffic Grooming Problem 26