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Genetic Algorithm for Multicast in WDM Networks
Der-Rong Din
Outline
Introduction Problem formulation Genetic Algorithm Further Research Problem
Introduction
There are two types of architectures of WDM optical networks: single-hop systems and multi-hop systems [2]. Single-hop system
a communication channel should use the same wavelength throughout the route of the channel
Multi-hop systema channel can consist of multiple light-paths and wavelength
conversion is allowed at the joint nodes of two light-paths in the channel.
In this paper, we consider single-hop systems, since all-optical wavelength conversion is still an immature and expensive technology. (no wavelength conversion)
Introduction
Multicast is a point to multipoint communication, by which a source node sends messages to multiple destination nodes.
A light-tree, as a point to multipoint extension of a light-path, is a tree in the physical topology and occupies the same wavelength in all fiber links in the tree.
Introduction
Each node of the tree is a multicast-Incapable optical switch (MI node) .
Introduction
The problem is formalized as follows: given an multicast request in a WDM netwo
rk system, compute a set of routing trees and assign wavelengths to them.
The objective is to minimize the (cost + α* # of wavelength) number of distinct wavelengths to be used
under the following constraints on each routing tree:
the total cost of the tree.
System Models
WDM network Connected and undirected graph G(V, E, c) V: vertex-set, |V|=n E: edge-set, |E|=m Each edge e in E is associated with a weight
function c(e): communication cost
System Models
Cost of path P(u,v):
A multicast request in the system are given, denoted by r (s, D) source s destination: D={d1, d2, ..., d|D|}
),(
)()),((vuPe
ecvuPc
System Models
This paper assumes an input optical signal can only be forward to an output signal at a switch.
Tk (s, Dk) be the routing tree for request r (s, D) in wavelength k, where k<K, T=∪ k=1,2,...,KTk;
D=∪ k=1,2,...,K Dk; T is the light-forest. The light signal is forwarded to the output port
leading to its child, which then transmit the signal to its child until all nodes in the Dk receive it.
Objective
The cost of the tree
where yj =1 if wavelength j is used; yj=0, otherwise Special case:
One objective of the multicast routing is to construct a routing tree (or forest) which has the minimal cost. The problem is regarded as the minimum Steiner tree problem, which was proved to be NP-hard.
Another objective is to minimize the number of wavelengths used in the system.
In a single-hop WDM system, two channels must use different wavelengths if their routes share a common link, which is the wavelength conflict rule.
y )()),((K
1jj
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DsTe i
ecDsTc
Genetic Algorithm for WDM Multicast Problem (WDMMP) Important components of GA
Chromosome encoding Fitness function Penalty function Crossover operation Mutation operation.
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r(s, {1,2,3,4,5,6}
Example of GA
since out-degree(s)=4, |D|=6, thus may be 2 wavelengths are need to multicast the request.
Genetic Algorithm #1
Basic idea: modified the GA of R-H Whang et al. to WDM network
pi is between 1 and Ri, i=1,2,...,|D|, where Ri is the number of candidate path from s to di
p1 p2 p3 p4 pi P|D|
p1 p2 p3 p4 pi P|D|
Chromosome Encoding
Light-Forest Construct Algorithm
Path by path construct Integrated the path and wavelength in single
phase Step 1: Sort paths in increasing order
according to the cost of each path O(|D| log |D|) time. Assume that p1,p2,...., p|D| be the new index.
Step 2: p1 is assigned to wavelength 1,w=1, T1={p1}, T2= ...=Tk=ø. O(n)
Light-Forest Construct Algorithm Step 3: For i= 2 to |D] do Begin
j=1 while j≦w do
{ if pi is not conflict with Tj
then {assigned pi to Tj Tj=Tj ∪pi flag=TRUE}
else j=j+1 } if flag is not TRUE
then w=w+1 Tw=Tw ∪ pi
End
Time complexity: O(|D|2*n)
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Example
p1=s7 1 (10)p2=s7 14 2 (13)p3=s9 13 3 (15)p4=s10 4 (8)p5=s10 4 5 (12)p6=s9 13 5 6 (26)
cost=8+10+4+15+13+26+2*α
Conflict Test Algorithm for path and Tree light-tree is represented by a directed tree
root at s. O(n) time: add path into a directed tree, then
test the out-degree of the visited vertex, if the out-degree >1 then conflict occurred.
Penalty Function
The light-forest construct a feasible solution of the WDM network, thus, there is no need for the penalty function.
Minimized
Transform to maximization form
where Cmax denotes the maximum value observer so far of the cost function in the population.
Fitness Function
Fitness =Cmax-Cost
Algorithm
W
jj
W
jj yTCost
11
)(cost
Crossover Operator
single point crossover multiple point crossover
Single point Crossover
2 3 1 4 1 3 1 2 2 3 2 1
2 3 1 4 2 1 1 2 2 3 1 3
After crossover, the light-forest should be reconstructed
Multiple point Crossover
2 3 1 4 1 3 1 2 2 3 2 1
2 3 2 3 1 3 1 2 1 4 2 1
After crossover, the light-forest should be reconstructed
Mutation Operator
single point mutation heuristic mutation
Single point mutation
After single point mutation, the light-forest may be changed.
The old path is traversed backward from di to s The edge we traversed are removed If the use
(e)=1 until the following saturations occurred, reach s reach destination node dl in D which p l is assigne
d to the same wavelength reach a node with out-degree > 1.
Example of single point mutation
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p1=s7 1 (10)p3=s9 13 3 (15)p4=s10 4 (8)p5=s10 4 5 (12)
Example of single point mutation
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p1=s7 1 (10)p3=s9 13 3 (15)p4=s10 4 (8)p5=s10 4 5 (12)
if p5 is mutated to p5=s85then the old path 4 5 is removedand new path is tested whether is conflict to current light-tree or not.if no then assign new path to current wavelength.otherwise, another light-tree ofdifferent wavelength is tested and selectedto assign.
Example of single point mutation
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p1=s7 1 (10)p3=s9 13 3 (15)p4=s10 4 (8)p5=s10 4 5 (12)
if p4 is mutated to p4=s1012 4then the old path 4 5 is not removedand new path is tested whether is conflict to current light-tree or not.if no then assign new path to current wavelength.otherwise, another light-tree ofdifferent wavelength is tested and selectedto assign.
Example of mutation
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Heuristic Mutations
Wavelength reduced mutation try to reduced the number of wavelengths use
d by the mutlicast request Cost reduced mutation
try to reduced the cost of each light-tree of different wavelengths used by the mutlicast request
Wavelength reduced mutation
Let number dest(wi) be the number of destination nodes in the wavelength wi.
Find out the minimal dest(wi) of paths. Wavelength reduced mutation is reassigned the desti
nation in this wavelength to another. Local optimal steategry.
Wavelength reduced mutation algorithm
For the destination di which is selected to be assigned to another wavelength, choose wavelength wk, k is initially set to be 1.
Remove the current light-tree in wavelength wk and form the graph G’, find a minimal cost path form s to G’, find minimal paths from dl to di, where dl is the destination n
ode in wavelength wk and is a leaf node, Find the minimal cost of these paths resulted from 1 and 2. Reassign the wavelength of path pi to wk, Change the chromosome encoding in pi field to correspondi
ng index.
Data structure
The operation of the “Change the chromosome encoding in pi field to corresponding index” may cause some problem The new search path from s to di may not included in the rati
ng table Ri. The searching time of path is long. To avoid the duplicated in the Ri, the operation should
check whether or not the new path has been included in the Ri,
if yes then return the corresponding index if no, then new path should be inserted into the Routing Tabl
e Ri of di, If the data structure of the routing table do not well-designed
then the time spent for the heuristic mutation will long.
Data structure
Operation: Given a index pi, return the path from s to di. Given a path, check that whether this is path is in the R
i, if yes return the index of pi; otherwise, insert this path into Ri, and return the new index of pi.
Data structure Index array (IA) Depth search tree (DST) Double Links between DST and IA
DST
For each destination di, Find k-shortest path for the di from s to di on G.
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DST
some paths from s to 6s 7 14 2 16 17 6s 7 14 2 15 6s 7 14 2 15 5 6s 7 14 2 11 3 13 5 6s 7 14 2 11 3 9 8 5 6s 7 14 2 11 3 13 1 9 8 5 6s 10 4 5 6s 10 12 5 6s 10 4 12 5 6
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Cost reduced mutation
For each wavelength (each ligth-tree), if dest(wi) >1 then fine the longest path in this light-tree, try to find another shorter path to replaced it. That is:
find a minimal cost path form s to G’, find minimal paths from dl to di, where dl is the destination n
ode in wavelength wk and is a leaf node, Find the minimal cost of these paths resulted from 1 and 2. Reassign the wavelength of path pi to wk, Change the chromosome encoding in pi field to correspondi
ng index.
Notice
The IA and DST structure were established during the initial phase.
Some Problem
The set of paths should be used to construct a tree of forest on WDM network to satisfy the wavelength constraint.
An tree constructing algorithm is needed. About O(|D|*n)
An wavelength assignment is needed. About O(e) time.
An integrated algorithm can be proposed to combine two algorithms.
Time complexity analysis
Random generated a population path-oriented gene without wavelength assignment.
Determine the result WDM-forest by applying integrated algorithm.
Time complexity: O(e + |D|n)* population_size * generation_size.
Paper Figure
IP router
WDM switch
s d1
d2
IP router
WDM switch
s d1
d2
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Example
p1=s7 1 (10)p2=s7 14 2 (13)p3=s9 13 3 (15)p4=s10 4 (8)p5=s10 4 5 (12)p6=s9 13 5 6 (26)
cost=8+10+4+15+13+26+2*α
Pair (s,di) path Cost
P1=(s,1) s7 1 10
P2=(s,2) s7 14 2 13
P3=(s,3) s9 13 3 15
P4=(s,4) s10 4 8
P5=(s,5) s10 4 5 12
P6=(s,6) s9 13 5 6 26
Example
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p1=s7 1 (10)p2=s7 14 2 (13)p3=s9 13 3 (15)p4=s10 4 (8)p5=s10 4 5 (12)p6=s9 13 5 6 (26)
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