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