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Distributed Computing Optical networks: switching cost and traffic grooming

Shmuel Zaks

zaks@cs.technion.ac.il

©

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OutlineOutline

Optical networksOptical networks ModelModel The Min ADM ProblemThe Min ADM Problem The Traffic Grooming The Traffic Grooming

ProblemProblem Algorithm GROOMBYSCAlgorithm GROOMBYSC

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the fiber serves as a transmission medium

Electronic switch

Optic fiber

Optical networks - 1st generation

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Routing in the optical domainTwo complementing technologies:- Wavelength Division Multiplexing (WDM):

Transmission of data simultaneously at multiple wavelengths over same fiber- Optical switches: the output port is determined according to the input port and the wavelength

Optical networks - 2nd generation

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Wavelength Division Multiplexing (WDM)

Directed:1234

Symmetric:1234

Undirected:1234

Optic Fiber

Optic Fiber

Optic Fiber

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Optical Switches

No two inputs with the same wavelength should be routed on the same edge.

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Lightpaths

ADM

ADM

Data in electronic form

Data in electronic form

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A virtual topology

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Lightpaths

p1

p2

1 2( ) ( )w p w p

Valid coloring

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The Routing Problem Input :

A graph G=(V,E) A set or sequence of node pairs (ai,bi)

Output: A set or sequence of paths pi =(ai, v1, …,

bi)

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The Load Given a graph G=(V,E) and a set P of

paths on the graph, we define: for any edge e of the graph:

the load on this edge l(e)=|Pe|

The (maximum, minimum, average) load on the network:

| eP p P e P

max

min

max ( ) |

min ( ) |

( ) avge E

L L l e e E

L l e e E

L l e E

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Wavelength Assignment Problem (WLA) Input:

A graph G=(V,E). A set or sequence of paths P.

Output: A coloring w of the paths:

Constraint::w P Na

, , ' , ' ( ) ( ')ee E p p P p p w p w p

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Routing and WLA (RLA/WRA)

Input : A graph G=(V,E) A set or sequence of node pairs (ai,bi)

Output: A set or sequence of paths pi =(ai, v1, …, bi) A coloring w of the paths: Constraint:

:w P N, , ' , ' ( ) ( ')ee E p p P p p w p w p

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Cost Measure: # of colors

For any legal coloring w of the paths:

Range( ) ( ) ( ) |W w w P w p p P

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Optimization Problems Goal:

MINW: Minimize W. or MAXPC: Maximize |Domain(w) | under

the constraint W<=Wmax.

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Static vs. Dynamic vs. Incremental

Static: The input is a set (of pairs or paths), the algorithm calculates its output based on the input.

Incremental (Online): The input is a sequence of input elements (pairs or

paths). It is supplied to the algorithm one element at a time. The output corresponding to the input element is

calculated w/o knowledge of the subsequent input elements.

Dynamic: Similar to incremental The sequence may contain deletion requests for

previous elements.

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WLA (A trivial lower bound) For any instance of the WLA

problem: W>=L.

Proof: Consider an edge e, such that L=l(e). There are L paths p1, …, p|L| using e,

because the paths are simple. Therefore :

, | |, ( ) ( )i ji j L w p w p

( ) |1iw p i L L

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WLA (A trivial lower bound) For some instances W > L.

L=2

W=3

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Static WLA in Line Graphs The GREEDY algorithm: // The set of integers for i = 1 to |V| do

for each path p=(x,i) do for each path p=(i,x) do

( )W W w p

{ ( ) min ;

\ ( )}

w p W

W W w p

W N

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Static WLA in Line Graphs Correctness, obvious. Optimality: By induction, After node i is processed, the

claim is correct, i.e. Where

W(i) is the value of after node i is pocessed, and

L(i) is the maximum load on the edges processed so far.

( ) ( )W i L i

max WN

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OutlineOutline

Optical networksOptical networks Model Model The Min ADM ProblemThe Min ADM Problem The Traffic Grooming The Traffic Grooming

ProblemProblem Algorithm GROOMBYSCAlgorithm GROOMBYSC

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Electronic ADM

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number of wavelengthsSwitching cost

ADM

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The MIN ADM Problem

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

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The Goal

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

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Static WLA in Line Graphs Note: After a slight modification, the Greedy

algorithm solves optimally the MINADM problem too:

At each node, first use the colors added to at this step.

It’s straigtforward to show that this: Does not harm the optimality w.r. to the MINW prb. Solves the MINADM problem optimally at each node.

W

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Static WLA in Line Graphs The GREEDY algorithm: // The set of integers for i = 1 to |V| do

for each path p=(x,i) do for each path p=(i,x) do

( )W W w p

{ ( ) min ;

\ ( )}

w p W

W W w p

W N

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W-ADM tradeoff

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

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ring (Eilam, Moran, Zaks, 2002) reduction from coloring of circular arc

graphs.

NP-complete

Minimizing # of ADMs –

Gerstel, Lin, Sasaki, 1998

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Coloring of Circular arc Graphs

Consider: a ring H (the host graph) and A set of paths P in H.

The graph G=(P,E) constructed as follows is a circular arc graph:

There is an edge (p1,p2) in e if and only if p1 and p2 have a common edge in H.

The problem of finding the chromatic number of a circular arc graph is NP-Hard [Tuc 75’]

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The reduction

The min W problem is exactly the circular arc coloring problem. But we will show NP-hardness even of the special case L=Lmin.

Given an instance C,P where C is the ring and P is the set of paths, we construct an instance C, P’ (by adding paths to P) such that Lmin(P’)=L(P’)=L(P).

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The reduction (cont’d) Claim: P is L-colorable iff P’ is L-

colorable.

Claim: ADM(P)=ADM(P’).

Therefore w.l.o.g. all the edges have the same load (L).

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|ADMs|=7=7+0

|ADMs|=9=6+3

|ADMs| = N + |chains|

Basic observationN lightpaths

cycles

chains

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The reduction (cont’d)

P’ is L-colorable iff

P’ can be partitioned into L cycles iff

ADM(P’)=|P’|.

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R ALG 2R R OPT 2R

ALG 2 x OPT

R: # of lightpaths ALG: # of ADMs used by the algorithm OPT: # of ADMs used by optimal solution

Approximation algorithms

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3/2 - Calinescu, Wan , 2002 10/7+ - Shalom, Z. , 2004 10/7 - Epstein, Levin, 2004

ALG 2 x OPT

Approximation algorithms

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OutlineOutline

Optical networksOptical networks Model Model The Min ADM ProblemThe Min ADM Problem The Traffic Grooming The Traffic Grooming

ProblemProblem Algorithm GROOMBYSCAlgorithm 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 R: The # of requests. SOL: The # of ADMs used by a

solution. OPT: The # of ADMs used by an

optimal solution.R/g SOL 2RR/g OPT 2RSOL = SOL/OPT 2g

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OutlineOutline

Optical networksOptical networks ModelModel The Min ADM ProblemThe Min ADM Problem The Traffic Grooming The Traffic Grooming

Problem Problem Algorithm GROOMBYSCAlgorithm 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

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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)) weightALG (SC)

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 OPT

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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 of OPT.Consider the set P of paths colored .Consider the set of ADMs operating at wavelength . (i.e. endpoints(P) )Divide endpoints(P) into sets of k consecutive nodes.For simplicity assume |endpoints(P)|=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

, ,, [ ] endpoints( ) .i ii weight S S k g

thus

,,

ii

P S

Moreover

,iSC S Therefore

Is a set cover considered by the algorithm.