1 distributed computing optical networks: switching cost and traffic grooming shmuel zaks...
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Distributed Computing Optical networks: switching cost and traffic grooming
Shmuel Zaks
©
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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.