Dynamic Wavelength Allocation in All-optical Ring Networks

Ori Gerstel and Shay Kutten

Proceedings of ICC'97

Static WLA in Rings

Definitions:

• For any we define:

• and :

•

•

v V{ | is an intermediate node of P}vP p P v

( ) | |node vl v P

min min{ ( ) | }node nodeL l v v V max max{ ( ) | }node nodeL l v v V

Static WLA in Rings

Algorithm:• Choose a node v such that.• Duplicate node v and cut the cycle to form a

line graph.• Color the paths in using the greedy

algorithm for line graphs with at most L colors.• Color the paths in using colors:

Number of colors used by the algorithm:

min( )node nodel v L

vP P

vP min 1nodeL L

2 1W L

Static WLA in Rings

v v

Static WLA in Rings

•L=4

•W=7

The above algorithm is optimal for some instances:

Dynamic Routing in Rings

• Input : A sequence of node pairs (si,ti).• Output : for each (si,ti) decide online,

“CLOCKWISE” or “COUNTER CLOCKWISE.”• Goal : minimize L.An online algorithm:

– Does not have any knowledge of subsequent inputs (j > i).

– Can not change its decision on previous input elements (j < i).

Shortest Path Routing

Algorithm SHORT:

Given a pair (si,ti) route it on the shortest path on the ring.

Claim: SHORT is 2 competitive ( )2SHORT OPTL L

Shortest Path Routing

Proof:

Consider an edge e such that :

Let

In OPT, there are at least x paths that use the edge e’ opposite to e.

Then:

( )SHORT SHORTl e L

( ) ( )SHORT OPTx l e l e

( ') ( ) ( )

( )

2

OPT OPT SHORT OPT

SHORT OPT SHORT OPT

SHORT OPT

L l e x l e l e

L l e L L

L L

Dynamic WLA in Rings

• Algorithm WLA-1(Lalg).

– It depends on an additional parameter which is the maximum anticipated load (L<=Lalg).

– Pools of 2 Lalg wavelengths each.

1. Given a path p, let l(p) its length. Choose i such that:

2. If the request is insert

3. If the request is delete

log N 0 1 log 1, ,..., NW W W

1( )

2 2 2i i

N Nl p

( ) min ; { ( )}i i iw p W W W w p

{ ( )}i iW W w p

Dynamic WLA in Rings

• Claim: as long as L<=Lalg, upon entering to step 2 .

• Corollary: the algorithm colors all the paths using at most wavelengths.

iW

alg2 logL N

Dynamic WLA in Rings

• Proof: Assume 1( )

2N l p N

A0

B0

Dynamic WLA in Rings

A0

B0

There are at most Lalg such paths traversing A0.

They can be colored using at most Lalg colors

The paths not traversing A0 ,do traverse B0. They can be colored using at most Lalg colors.

We use at most 2 Lalg colors for paths from class 0.

Dynamic WLA in Rings

AiThere are two types of paths:

•Paths traversing (exactly) one Ai edge, (A)

•Paths traversing no Ai edge, (B). These edges traverse exactly one B edge. Ai

Bi

Bi

We have two sets of L colors for each of the A and B paths.

Dynamic WLA in Rings

• Algorithm WLA-2.– Same as WLA-1, except…– The pools are not static. We have a global

pool of wavelengths.

– As long as L<=Lalg,

• Algorithm WLA-3– No a priori allocation.–

alg2 logL N

alg2 logW L N

2 logW L N

Dynamic WLA in Rings

• A lower bound:– Assume L=2– We describe an adversary, which works in

phases.– Phase i ends when the algorithm uses i

wavelengths.– At each phase the adversary issues requests

of length at most 2i.

– There are phases. Therefore:log N log 0.5 logW N L N

Dynamic WLA in Rings

Dynamic WLA in Rings

•For any even L, the above adversary will issue L/2 requests instead one.

•We get again the same lower bound:

1log

2W L N