Dynamic Wavelength Allocation
and Wavelength Conversion
Wavelength Converters
• A wavelength converter is modeled by a bipartite
graph.
• For any two adjacent edges x,y, we define the
corresponding conversion graph Gxy=(Vx,Vy,Exy)
where:
• Vx={x0,x1,…xW-1}
• Vy={y0,y1,…yW-1}
• ( xi , y j )  Exy if and only if wavelength i on
endpoint x can be converted to wavelength j on
endpoint y (i is compatible with j on x-y ).
Wavelength Converters
• A wavelength converter is symmetric if:
(i, j )  Exy  ( j, i)  Exy
• A full wavelength converter corresponds to
a complete bipartite graph.
• The degree of a wavelength converter is the
maximum degree of a node in the bipartite
graph.
Full Wavelength Converters
• Any instance can be colored using W=L
wavelengths. (In fact we need only d>=L)
• Proof:
– Given a path p  P, we can color its edges
independently of each other, because the full
conversion capability.
– Consider any edge e  p : There are at most
L-1 paths other than p traversing this edge.
They use at most L-1=W-1 colors. We can use
any one from the remaining colors for p.
Expander Graphs
• Definition: Given any S  V , we define the
neighborhood of S, namely:
( S )  {v V | u  S , (u, v)  E}
• Definition: A bipartite graph (V1,V2, E) is an
(a, b, d)-expander if:
– each node has degree at most d.
–0<a<½
b>1
– for any S  V1 , S  a V  (S )  b S
Expander Graphs
• Lemma: There is a triple (a,b,d), such that:
for every sufficiently large n, there is an
(a,b,d)-expander with n nodes.
Limited Wavelength Converters
(Any Graph)
• Theorem: There exists two constants k>1 and
d>1, such that every instance can be colored with
– W=kL colors
– Using wavelength converters with degree d.
• Proof: Between each two adjacent edges we use
the converter which correspond to the (a,b,d)expander whose existence is guaranteed by the
previous lemma.
• Let k=1/min{a(b-1),1-a}
• We will prove that as long as L <= W min{a(b1),1-a}, any path can be colored.
Limited Wavelength Converters
(Any Graph)
• Assume L <= W a(b-1) and L<=W(1-a)
• Consider a path p=(e1, e2, …, el) to be
colored.
• For any edge e1, e2, …, el a color is said to
be busy if it is used by another path.
• For any edge ei, (i>1) a color c is said to
be busy also if all the colors compatible
with c are busy in ei-1
Limited Wavelength Converters
(Any Graph)
• Claim:There are aW colors which are not
busy (idle) in ei. (By induction on i)
– i=1:L<=W(1-a), therefore there are aW colors
idle in e1.
– i > 1:
• In edge ei+1 there are at least baW colors
compatible with the idle colors of ei
• At most L<W a(b-1) of them are used by other
lightpaths.
• We are left with at least baW- Wa(b-1)= a W idle
colors.
Limited Wavelength Converters
(Rings)
• Theorem: Any instance of ring graph can be
colored with
– W=L log L + 4 L colors (independent of N !!)
– using converters of degree 2.
• Proof:
– Divide the ring into segments of length at least L, but
less than 2L.
• Wline(N,L)<=L log N (prove)
• Wline(2L,L)<=L log N + L
• We can color the intra-segment paths with L logN + L colors
with no wavelength conversion.
Limited Wavelength Converters
(Rings)
– Use the following graph to color inter-segment
paths:
• An edge of the graph is a color.
• A vertex joins compatible colors.
u1
u2
uL
First segment
v1
v2
vL
Intermediate segment
Last segment
Incremental WLA in Rings
• Claim: Any instance in Ring graphs can be
colored using W <= max{L, 2L-d}colors.
• Algorithm:
– Initialization:
•
•
•
•
•
M = max {0, L-d}
for i=0 to M do SHELF (i )  
POOL(0)={1,…,min{L, d}}
w=d
for i=1 to M do POOL(i)={++w, ++w}
Incremental WLA in Rings
Notation:
Si 
i
SHELF ( j )
j 0
• l(e/S) -Load induced on edge e by paths in S,
namely:  p  S | e  p
• Note that l(e)=l(e/P).
• l ( p / S )  max{l (e / S ) | e  p}
• Fi the set of paths received before path i, namely:
Fi={p1,p2,…,pi-1}
Incremental WLA in Rings
• Algorithm (path p)
–
–
–
–
i=0;
While L(p/Si) >= d+i do i++
SHELF (i )  SHELF (i ) { p}
Color the edges of p using wavelengths from
SHELF(i)
Incremental WLA in Rings
Lemma:
Let x  y, i  0, px p y  ,{ px , p y }  SHELF (i)
then L( px p y / Fy Ti 1 )  d  i  1
Proof: Assume L( px p y / Fy Ti 1 )  d  i  1
p  T  T and p  F , therefore
x
L( px
i
i 1
p y / Fy
x
y
Ti )  d  i
contradicting to the fact that
p y  SHELF (i )
.
Incremental WLA in Rings
Lemma: Let i  0,{ px , p y }  SHELF (i)
then px  p y , p y  px
Proof: w.l.o.g. x<y.
• Assume p  p , then by previous lemma
y
L( p y / Fy
x
Ti 1 )  d  i  1
The algorithm would place py in
SHELF(j) for some j<i.
Incremental WLA in Rings
• Assume
px  p y
px  SHELF (i)  SHELF (i  1)
L( px / Fx
Ti 1 )  d  i  1
L( p y / Fy
Ti 1 )  d  i
, therefore
The algorithm would place py in SHELF(j)
for some j>i.
Incremental WLA in Rings
• The maximum load induced by the paths
of SHELF(0) is d. By code inspection.
• The maximum load induced by the paths
of SHELF(i) (i>0) is 2.
– Assume otherwise. There is an edge with three
paths p , p , p  SHELF (i) traversing it. By previous
lemma, none of them contains the other.
W.l.o.g assume they are sorted by their starting
points:
x
y
z
py
px
pz
Incremental WLA in Rings
By the above picture, for any set S of paths:
L( p y / S )  max( L( px
p y / S ), L( px
p y / S ))
By the first lemma: L( px
p y / Fmax( x, y )
Ti 1 )  d  i  1
L( pz
p y / Fmax( z , y )
Ti 1 )  d  i  1
Load is non decreasing:
L( px
p y / Fy
Ti 1 )  L( px
p y / Fmax( x , y )
Ti 1 )
Ti 1 )  L( pz p y / Fmax( z , y ) Ti 1 )
Combining, we get: L( p y / Fy Ti 1 )  d  i  1
L( pz
p y / Fy
The algorithm would place py in SHELF(j) for some
j<i.