© Distributed Computing
Optical networks: switching
cost and traffic grooming
Shmuel Zaks
[email protected]
1
Outline

Optical networks
 Model
 The Min ADM Problem
 The Traffic Grooming
Problem
 Algorithm GROOMBYSC
2
Optical networks - 1st generation
the fiber serves as a transmission medium
Electronic
switch
Optic
fiber
3
Optical networks - 2nd generation
Routing in the optical domain
Two 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
4
Wavelength Division
Multiplexing (WDM)
4
3
2
1
Directed:
Optic Fiber
4
3
2
1
Symmetric:
Optic Fiber
4
3
2
1
Undirected:
Optic Fiber
5
Optical Switches
No two inputs with the same wavelength should be
routed on the same edge.
6
Lightpaths
ADM
ADM
Data in
electronic
form
Data in
electronic
form
7
A virtual
topology
8
Lightpaths
p1
Valid coloring
w( p1 )  w( p2 )
p2
9
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)
10
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:



Pe   p  P | e  P
the load on this edge l(e)=|Pe|
The (maximum, minimum, average) load on
the network:
L  Lmax  max l (e) | e  E
Lmin  min l (e) | e  E
Lavg   l (e)
eE
E
11
Wavelength Assignment
Problem (WLA)

Input:



Output:


A graph G=(V,E).
A set or sequence of paths P.
A coloring w of the paths: w : P a N
Constraint:
e  E, p, p '  Pe , p  p '  w( p)  w( p ')
12
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:
w: P N
Constraint: e  E, p, p '  P , p  p '  w( p)  w( p ')
e
13
Cost Measure: # of colors

For any legal coloring w of the paths:
W  Range( w)  w( P)  w( p) | p  P
14
Optimization Problems

Goal:

MINW: Minimize W.
or

MAXPC: Maximize |Domain(w) | under the
constraint W<=Wmax.
15
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.
16
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.
i, j | L |, w( pi )  w( p j )
Therefore :
w( pi ) |1  i  L  L
17
WLA (A trivial lower bound)

For some instances W > L.
L=2
W=3
18
Static WLA in Paths



The GREEDY algorithm:
W  N // The set of integers
for i = 1 to |V| do


for each path p=(x,i) do W  W w( p)
for each path p=(i,x) do { w( p)  min W ;
W  W \ w( p )}
19
Static WLA in Line Graphs




Correctness, obvious.
Optimality:
By induction, After node i is processed, the claim is
correct, i.e. W (i )  L(i )
Where


W(i) is the value of max N  W
after node i is
pocessed, and
L(i) is the maximum load on the edges processed so far.
20
Outline

Optical networks
 Model
 The Min ADM Problem
 The Traffic Grooming
Problem
 Algorithm GROOMBYSC
21
Electronic ADM
22
Switching
number
of cost
wavelengths
ADM
23
The MIN ADM Problem
W=2, ADM=4
W=1, ADM=3
24
The Goal
Given a set of lightpaths, find a valid
coloring with minimum number of
ADMs.
25
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 W 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.
26
Static WLA in Line Graphs



The GREEDY algorithm:
W  N // The set of integers
for i = 1 to |V| do


for each path p=(x,i) do W  W w( p)
for each path p=(i,x) do { w( p)  min W ;
W  W \ w( p )}
27
W-ADM tradeoff
W=2, ADM=8
W=3, ADM=7
28
NP-complete
Minimizing # of ADMs –
Gerstel, Lin, Sasaki, 1998
ring
(Eilam, Moran, Zaks, 2002)
reduction from coloring of circular arc
graphs.
29
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’]
30
The special case k=L=Lmin



The min W problem is exactly the circular arc
coloring problem. But we will show NPhardness even of the special case k=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 of unit length to P)
such that Lmin(P’)=L(P’)=L(P)=k.
Claim: P is k-colorable iff P’ is k-colorable.
31
Basic observation
R lightpaths
cycles
chains
|ADMs|=9=6+3
|ADMs| = R + |chains|
|ADMs|=7=7+0
32
NP-Hardness of Min-ADM



Given an instance C,P where C is the ring and
P is the set of paths, and Lmin(P)=L(P)
P is L(P)-colorable iff it can be partitioned into
L(P) rings.
P can be partitioned into L(P) rings iff ADM(P)
= |P|
33
Approximation algorithms



R: # of lightpaths
ALG: # of ADMs used by the algorithm
OPT: # of ADMs used by optimal solution
R  ALG  2R
R  OPT  2R
ALG  2 x OPT
34
Approximation algorithms
ALG 
2 x OPT
3/2 - Calinescu, Wan , 2002
10/7+ - Shalom, Z. , 2004
10/7 - Epstein, Levin, 2004
35
Outline

Optical networks
 Model
 The Min ADM Problem
 The Traffic Grooming
Problem
 Algorithm GROOMBYSC
36
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.
37
The Traffic Grooming
Problem
g=2
W=2, ADM=8
W=1, ADM=7
38
The Goal
Given a set of requests and a grooming
factor g, find a valid coloring with
minimum number of ADMs.
39
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  2R
R/g  OPT  2R
rSOL = SOL/OPT  2g

40
Outline

Optical networks
 Model
 The Min ADM Problem
 The Traffic Grooming
Problem
 Algorithm GROOMBYSC
41
Main Result
g > 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.
42
Approximation algorithm (log g)
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 £ k ×g
If a subset A is 1-colorable (i.e., any edge is
used at most g times) then
weight[A]=endpoints(A);
S ¬ S È {A}
43
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
44
Analysis
Let
A B
, then:
If B is 1-colorable then A is 1-colorable
(correctness).
Cost(A)  Cost(B).
Therefore: …
45
ALG = cost(PARTITION)
 weight(COVER) 
H
k g
weight(MINCOVER)
 (1 +ln(k  g))w eight(SC)
for every set cover SC.
46
ALG  (1 +ln(k  g)) weight (SC)
for any set cover SC.
Lemma: There is a set cover SC, s.t.:
 2g 
weight(SC)   1 +
O PT

k 

47
Conclusion:
ALG  weight(COVER) 
H
k g
weight(MINCOVER)
 (1 +ln(k  g))weight(SC) 
 2g 
(1 +ln(k  g))   1 +   OPT
k 

For k = g ln g : A L G   2lng + o(lng )   OPT
48
Proof of Lemma
Lemma: There is a set cover SC, s.t.:
 2g 
weight(SC)   1 +
O PT

k 

49
Proof of Lemma
Consider 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
50
M=4
k=6
k
S1
k
k
k
S2
Sm
weight[S1 ]  k + g
51
Analysis (cont’d)
weight[ Si , ]  k  g
m
 weight[S  ]  m(k  g )
i 1
i,
OPT  m.k
 g
weight[ Si , ]  OPT 1  

 k
i 1
m
w/o the assumption we have:
 2g 
weight[ S i , ]  OPT  1 


k


i 1
m
 2g 
weight[ S i , ]  OPT  1 
 

k


i 1
m
52
Analysis (cont’d)
 i ,  weight [ S i ,  ]  endpoints( S i ,  )  k . g
S i ,  S
Therefore
SC   S i , 
thus
S i ,  S
Moreover
P
i ,
S i ,
Is a set cover considered by
the algorithm.
53