Genetic Algorithm for Multicast in WDM Networks

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Genetic Algorithm for Multicast in WDM Networks

• Der-Rong Din

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Outline

• Introduction
• Problem formulation
• Genetic Algorithm
• Further Research Problem

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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 system
• a 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)

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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.

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Introduction

• Each node of the tree is a multicast-Incapable
  optical switch (MI node) .

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Introduction

• The problem is formalized as follows
• given an multicast request in a WDM network
  system, compute a set of routing trees and assign
  wavelengths to them.
• The objective is to minimize the (cost a of
  wavelength)
• number of distinct wavelengths to be used under
  the following constraints on each routing tree
• the total cost of the tree.

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System Models

• WDM network
• Connected and undirected graph G(V, E, c)
• V vertex-set, Vn
• E edge-set, Em
• Each edge e in E is associated with a weight
  function
• c(e) communication cost

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System Models

• Cost of path P(u,v)
• A multicast request in the system are given,
  denoted by r (s, D)
• source s
• destination Dd1, d2, ..., dD

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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 kltK, T?
  k1,2,...,KTk
• D? k1,2,...,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.

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Objective

• The cost of the tree
• where yj 1 if wavelength j is used yj0,
  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.

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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
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Example of GA

• since out-degree(s)4, D6, thus may be 2
  wavelengths are need to multicast the request.

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Genetic Algorithm 1

• Basic idea modified the GA of R-H Whang et al.
  to WDM network

pi is between 1 and Ri, i1,2,...,D, where Ri
is the number of candidate path from s to di
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Chromosome Encoding
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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,...., pD be the new index.
• Step 2 p1 is assigned to wavelength 1,w1,
  T1p1, T2 ...Tkø. O(n)

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Light-Forest Construct Algorithm

• Step 3 For i 2 to D do
• Begin
• j1
• while j?w do
• if pi is not conflict with Tj
• then
• assigned pi to Tj
• TjTj ?pi
• flagTRUE
• else jj1
• if flag is not TRUE
• then
• ww1
• TwTw ? pi
• End

Time complexity O(D2n)
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Example
p1s?7 ?1 (10) p2s?7 ?14 ?2 (13) p3s?9 ?13 ?3
(15) p4s?10 ?4 (8) p5s?10 ?4 ?5 (12) p6s?9
?13 ?5 ?6 (26)
cost81041513262a
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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 gt1 then conflict occurred.

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Penalty Function

• The light-forest construct a feasible solution of
  the WDM network, thus, there is no need for the
  penalty function.

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Fitness Function
Algorithm

• Minimized
• Transform to maximization form
• where Cmax denotes the maximum value observer so
  far of the cost function in the population.

Fitness Cmax-Cost
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Crossover Operator

• single point crossover
• multiple point crossover

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Single point Crossover
After crossover, the light-forest should be
reconstructed
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Multiple point Crossover
After crossover, the light-forest should be
reconstructed
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Mutation Operator

• single point mutation
• heuristic mutation

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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 pl is
  assigned to the same wavelength
• reach a node with out-degree gt 1.

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Example of single point mutation
p1s?7 ?1 (10) p3s?9 ?13 ?3 (15) p4s?10 ?4
(8) p5s?10 ?4 ?5 (12)
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Example of single point mutation
p1s?7 ?1 (10) p3s?9 ?13 ?3 (15) p4s?10 ?4
(8) p5s?10 ?4 ?5 (12)
if p5 is mutated to p5s?8?5 then the old path 4
?5 is removed and 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
of different wavelength is tested and selected to
assign.
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Example of single point mutation
p1s?7 ?1 (10) p3s?9 ?13 ?3 (15) p4s?10 ?4
(8) p5s?10 ?4 ?5 (12)
if p4 is mutated to p4s?10?12 ?4 then the old
path 4 ?5 is not removed and 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
of different wavelength is tested and selected to
assign.
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Example of mutation
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Heuristic Mutations

• Wavelength reduced mutation
• try to reduced the number of wavelengths used by
  the mutlicast request
• Cost reduced mutation
• try to reduced the cost of each light-tree of
  different wavelengths used by the mutlicast
  request

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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
  destination in this wavelength to another.
• Local optimal steategry.

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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 node 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
  corresponding index.

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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 rating 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 Table 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.

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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 Ri, 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

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DST

• For each destination di,
• Find k-shortest path for the di from s to di on
  G.

some paths from s to 6 s ?7 ?14 ?2 ?16 ?17 ?6 s
?7 ?14 ?2 ?15 ?6 s ?7 ?14 ?2 ?15 ?5 ?6 s ?7 ?14
?2 ?11 ?3 ?13 ?5 ?6 s ?7 ?14 ?2 ?11 ?3 ?9 ?8 ?5
?6 s ?7 ?14 ?2 ?11 ?3 ?13 ?1 ?9 ?8 ?5 ?6 s ?10 ?4
?5 ?6 s ?10 ?12 ?5 ?6 s ?10 ?4 ?12 ?5 ?6
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DST
some paths from s to 6 s ?7 ?14 ?2 ?16 ?17 ?6 s
?7 ?14 ?2 ?15 ?6 s ?7 ?14 ?2 ?15 ?5 ?6 s ?7 ?14
?2 ?11 ?3 ?13 ?5 ?6 s ?7 ?14 ?2 ?11 ?3 ?9 ?8 ?5
?6 s ?7 ?14 ?2 ?11 ?3 ?13 ?1 ?9 ?8 ?5 ?6 s ?10 ?4
?5 ?6 s ?10 ?12 ?5 ?6 s ?10 ?4 ?12 ?5 ?6
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IA DST
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Cost reduced mutation

• For each wavelength (each ligth-tree), if
  dest(wi) gt1 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 node 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
  corresponding index.

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Notice

• The IA and DST structure were established during
  the initial phase.

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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(Dn)
• An wavelength assignment is needed.
• About O(e) time.
• An integrated algorithm can be proposed to
  combine two algorithms.

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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 Dn) population_size
  generation_size.

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Paper Figure
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Example
p1s?7 ?1 (10) p2s?7 ?14 ?2 (13) p3s?9 ?13 ?3
(15) p4s?10 ?4 (8) p5s?10 ?4 ?5 (12) p6s?9
?13 ?5 ?6 (26)
cost81041513262a
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Example
p1s?7 ?1 (10) p2s?7 ?14 ?2 (13) p3s?9 ?13 ?3
(15) p4s?10 ?4 (8) p5s?10 ?4 ?5 (12) p6s?9
?13 ?5 ?6 (26)
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s-gt