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) .
• End-to-end delay is an important
  quality-of-service (QoS) parameter in data
  communications.
• QoS multicast requires that the delay of messages
  from the source to any destination be within a
  bound.

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Introduction

• The problem is formalized as follows
• given a multicast request in a WDM network
  system, compute a set of routing trees
  (light-trees) and assign wavelengths to them.
• The objective is to minimize the number of
  distinct wavelengths to be used under the
  following constraints on each routing tree
• the delay from the source to any destination
  along the tree does not exceed a given bound
• the total cost of the tree is suboptimal.
• or minimize cost awavelength

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System Models

• WDM network
• Connected and undirected graph G(V, E, c, d)
• V vertex-set, Vn
• E edge-set, Em
• Each edge e in E is associated with two weight
  functions
• c(e) communication cost
• d(e) the delay of e ( include switch and
  propagation delays)

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System Models

• Cost of path P(u,v)
• Delay of path P(u,v)
• A multicast request in the system are given,
  denoted by
• multicast request r (s, D, ?)
• source s
• destination D
• delay bound ?
• the data transmission delay from s to any node in
  D should be within bound ?

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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, ?k) be the routing tree for request r
  (s, D, ?) in wavelength k, where kltK, T?
  k1,2,...,KTk
• D? k1,2,...,K Dk ? max?k,k1,2,...,K, 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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QoS requirement

• The QoS requirement of routing tree Ti (s, Di,
  ?) is that the delay from s to any nodes in Di
  should not exceed ?.
• Let PTi(s, u) denotes the path in Ti (si, Di,
  ?i) from si to u in Di
• Thus,
• Assume
• where PG(s, u) is the shortest path s to u in G.

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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 RWA

• Important components of GA
• Chromosome encoding
• Fitness function
• Penalty function
• Crossover operation
• Mutation operation
• First, we assume rk1, for all k in K.

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Example of GA
connection request are (1,2), (1,4), (1,5),
(2,3), (2,4), (2,5), (3,4), (3,1), (4,2), (4,3),
(5,2), (5,4).
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Chromosome Encoding

• Wavelength gene yw binary

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Chromosome Encoding

• Connection gene ci, i1,...,M
• M number of connections. M12

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Example of connection gene
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Relation between two genes
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Objective Function

• Objective function
• The assignment represented by the connection may
  not constraint-satisfy, thus, a penalty function
  should be included in objective function.

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Penalty Function

• Assume both connections c1(1,2) and c2(1,4) are
  assigned to wavelength 1 with clockwise
  direction, then conflict occurred.
• Penalty should be defined.
• How to detect the conflict in a connection gene?
• A conflict-detection algorithm should be
  developed.
• O(M2) pairs of connections should be examined.
• The conflict between two connections can be
  detected in constant time O(1).

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Conflict-detection Algorithm

• Connections
• c1(1,4) ,
• c2(2,4) ,
• c1(1,2) ,
• c1(5,2)
• Construct four bipartite graph AA, AB, BA, BB,
• Node connection
• Edge conflict occurred
• A clockwise direction
• B counter-clockwise direction

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Graph AA
c4
c3
c1
c2

• c1(1,4) , c2(2,4) , c3(1,2) , c4(5,2)

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Graph AB
c4
c2
c3
c1
c1
c4
c2
c3

• c1(1,4) , c2(2,4) , c3(1,2) , c4(5,2)

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Graph BA
c4
c2
c3
c1
c1
c4
c2
c3

• c1(1,4) , c2(2,4) , c3(1,2) , c4(5,2)

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Graph BB

• c1(1,4) , c2(2,4) , c3(1,2) , c4(5,2)

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Lemma

• Lemma 1 AA and BB is symmetric
• Lemma 2 AB BAT
• Lemma 3 the time complexity of constructing
  graphs AA, AB, BA, BB are O(M2)
• Lemma 4 If AA, AB, BA, BB are available, the
  conflict of two connection can be known in O(1)
  time.

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Penalty function

• For a pair of connections (ci, cj) in connection
  gene, if conflict then associated a penalty a.
• Conflict means two connections use same
  wavelength and routing collision.

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Conflict
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Objective 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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Genetic crossover operator
Algorithm

• single point crossover an integer value i is
  generated in the range (1,M)

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Genetic crossover operator
Algorithm

• multiple points crossover k integer value i1,
  i2, ...,ik is generated in the range (1,M)

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Genetic operator
Algorithm

• global reverse routing operator each ci is
  multiplied (-1)

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Genetic operator
Algorithm

• partial reverse routing operator randomly select
  k (1?k ?M, integer) connections each ci is
  multiplied (-1) (e.g. k3)

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Genetic operator

• wavelength change operator randomly select two
  connections and the assigned wavelengths are
  exchanged.

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Mutation operator GA
Algorithm

• Random single connection mutation randomly a
  connection ci, mutate to a random selected
  wavelength and direction

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Mutation of GA

• Random multiple connection mutation randomly k
  connections ci, c2,..., ck mutate to random
  selected wavelengths and directions

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Mutation of GA

• Conflict-free mutation For all pairs of
  connections (ci, cj), find a set CF of pairs of
  connections, which are conflict-free in AA. AB,
  BA, BB.
• Rules
• If (ci, cj) in AA is conflict-free then (i, j) is
  added into CF.
• If (ci, cj) in AB is conflict-free then (i, -j)
  is added into CF.
• If (ci, cj) in AA is conflict-free then (-i, j)
  is added into CF.
• If (ci, cj) in AA is conflict-free then (-i, -j)
  is added into CF.

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Example of CF
CFAA(2,3), (2,4), (3,2), (4,2)
CFAB(2,-1), (2,-3), (3,-1), (3,-4)
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Example of CF
CFBA(-1,2), (-1,2), (-3,2), (-4,3)
CFBB(-1,-3), (-3,-1), (-3,-4), (-4,-3)
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Mutation of GA

• Assign each pair in CF an positive number
  increasing from 1 to CF.
• Randomly select a number, get the pair (i,j) of
  connections from CF corresponding to the number.
• Rules
• If igt 0 and jgt0 then mutate (ci to cj and cj
  to cj) or (cj to ci and ci to ci )
• If igt 0 and jlt0 then mutate (ci to -ci and ci
  to ci) or (ci to ci and cj to -ci )
• If ilt 0 and jgt0 then mutate (cj to ci and ci
  to -ci) or (ci to -cj and cj to cj )
• If ilt 0 and jlt0 then mutate (cj to -ci and ci
  to -ci) or (ci to -cj and cj to -cj )

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Example of Conflict-free Mutation

• CFCFAA?CFAB ? CFBA ? CFBB
• (2,3), (2,4), (3,2), (4,2), (2,-1), (2,-3),
  (3,-1), (3,-4), (-1,2), (-1,2), (-3,2), (-4,3),
  (-1,-3), (-3,-1), (-3,-4), (-4,-3)
• e.g. (-3,2) (-1, -3)
  (3,-4)

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Extended GA to handle rk gt1
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Modification of Graph AA
c4
c3
c1
c2

• c1(1,4) , c2(2,4) , c3(1,2) , c4(5,2)

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No modification in Graph AB and BA
c4
c2
c3
c1
c1
c4
c2
c3

• c1(1,4) , c2(2,4) , c3(1,2) , c4(5,2)

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Modification of Graph BB

• c1(1,4) , c2(2,4) , c3(1,2) , c4(5,2)

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Objective Function
where the index of connection gene in transformed
to 1,2,...,M
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Experiments
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Further Research

• Use Simulated Annealing Algorithm to solve this
  problem.
• Use GA and SA to solve another static version of
  optimization problem in WDM environment.

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Routing table of s ? 1 Routing table of s ? 1 Routing table of s ? 1
Path No. Path lists cost
1 s?7 ?1 10
2 s ? 7? 14 ? 1 11
3 s ? 9? 1 13
4 s ? 8 ? 9? 1 22
... ...
R
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p1 p2 p3 p4 pi PD
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p1 p2 ... pi ... pD w1 w2 ... wi wD