Optimal Placement of Wavelength Converters
in WDM Networks
X. Jia, D. Du, X.Hu, and D. Li
Dept of Computer Science,
City University of Hong Kong
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Wavelength Converters
Two major purposes of using wavelength converters:
 Reducing the blockings of channels in the system;
 Reducing the number of wavelengths required in the system.
Fig.
2
Wavelength Assignment and Traffic Load
Def. 1. Link Load: the traffic load over a link is the number of channels
over the link.
Def. 2. Maximal Link Load (System Load): the load of a system is the
maximal load over a link.
Notice:
1. The channels over the same link must be assigned with different
wavelengths.
2. The number of wavelengths W required in the system is at least
equal to the maximal link load L.
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Min # Wavelengths for a Given Load (NO Converters used)
Given the system load L, it is NP-hard to assign wavelengths to a set of
channels such that the number of wavelengths used is minimal. Some
results on Tree and Ring networks are given below:
Tree networks
Unidirectional channels
5
L wavelengths are sufficient
3
Duplex channels
3
L wavelengths are sufficient
2
5
4
3
2
L wavelengths are necessary
L wavelengths are necessary
Ring networks
(2L  1) wavelengths are both sufficient and necessary.
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Wavelength-Load Assignability (with Converter Used)
Def. 3. Given a network G(V,E), a set of nodes S, S  V, is said to
achieve load-wavelength assignability if, by equipping each node in S
with a wavelength converter, the number of wavelengths required for all
channels can be made equal to the maximal link load in the system, i.e.,
W = L.
Def. 4. Optimal Placement of Converters: find the minimal set of S for a
network G(V,E) such that W = L is achieved.
Exiting Results
Unidirectional channels
path
No converter is needed
star
No converter is needed
ring
1 converter is needed
5
Duplex channels
No converter is needed
1 converter is needed
1 converter is needed
Problem Specification
Our problem is to find the optimal placement of converters in a general
network graph G(V,E).
Our approach is to split a general network graph into paths and stars,
whose solutions are known.
Splitting operation: Given graph G(V,E) and a subset S  V, GS(V,E)
denotes the graph by splitting each node s  S into (s) nodes, one for
each edge incident to it in G(V,E).
Fig. 1
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Optimal Placement of Converters for Duplex Channels
Lemma 1. Given graph G(V,E), subset S  V achieves W = L if and only
if every connected component of GS(V,E) is a path.
Theorem 1. The OPC problem for duplex channels can be solved in time
of O(|E|+|V|).
Fig2 & fig3(a)
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Optimal Placement of Converters for Unidirectional Channels
Lemma 2. Given graph G(V,E), subset S  V achieves W = L if and only
if each connected component in GS(V,E) is a spider.
 Lemma 2 tells what S consists of. However, it is NP-complete to find
the minimal sized S that achieves W = L.
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Lemma 3. If graph G(V,E) has a node with degree greater than two, then
there exists a minimal sized subset S of V that achieves W = L and every
node in S has degree greater than two in G(V,E).
 Lemma 3 allows us to eliminate all nodes whose degree is less than
or equal to two in G(V,E) from consideration when searching for the
minimal sized S.
 Even graph G(V,E) is reduced to another graph G'(V,E) by removing
from G all the nodes having degree less than or equal to two, it is
still NP-hard to find the minimal sized S in G'.
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Lemma 4. If every node in G(V,E) has degree greater than two, then S
 V achieves W = L if and only if S is a vertex cover of G(V,E).
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An Illustration of the Algorithm
Fig.8
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The Algorithm
Input: G(V,E)
Output: C  V and C makes W = L
Step 1. process simple cases
If G(V,E) is a path then
return C = ;
If G(V,E) is a ring then
return C = any-node in V;
If G(V,E) is a spider then
return C = node whose degree  2;
Step 2. remove degree-two nodes;
Step 3. remove degree-one nodes;
Step 4. generate a vertex cover C of reduced
graph and return C.
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Theorem 2. The proposed algorithm can produce a set that achieves W =
L for network G(V,E).
Theorem 3. The proposed algorithm can produce a set C in time of
O(|E|+|V|) satisfying |C|  2 |S|, where S is the optimal solution to the
placement of converter problem.
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Conclusion and Future Work
 Optimal placement of converters uses the least number of converters
to achieve the best usage of wavelengths.
 Placement of converters to achieve W = L.
 Quantitative analysis of relationships between the placement of
converters and the system blocking probabilities in general network
graphs.
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