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Multicasting in a Class Multicasting in a Class of Multicast-Capable of Multicast-Capable WDM NetworksWDM Networks

From:From: Y. Wang and Y. Yang, Journal of Lightwave Y. Wang and Y. Yang, Journal of Lightwave

Technology, vol. 20, No. 3, Mar. 2002 Technology, vol. 20, No. 3, Mar. 2002

22

AbstractAbstract

Study Study multicastmulticast communication in a class communication in a class of of multicast-capablemulticast-capable WDMWDM networks (i.e., networks (i.e., have have light splitting switcheslight splitting switches) with regular ) with regular topologies under some commonly used rtopologies under some commonly used routing algorithms.outing algorithms.

Upper and lower boundsUpper and lower bounds on theon the minimum minimum number of wavelengthsnumber of wavelengths required are dete required are determined for a network to be rmined for a network to be rearrangeablerearrangeable for arbitrary multicast assignments.for arbitrary multicast assignments.

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OutlineOutline

IntroductionIntroduction Multicast-Capable WDM NetworksMulticast-Capable WDM Networks Generalization of the conflict graphGeneralization of the conflict graph RingsRings MeshesMeshes HypercubesHypercubes Linear ArraysLinear Arrays Comparison and ConclusionsComparison and Conclusions

44

IntroductionIntroduction

WDMWDM MulticastMulticast MulticastMulticast can be supported more can be supported more

efficiently in optical domain by efficiently in optical domain by utilizing the inherent utilizing the inherent light light splittingsplitting capacity of optical capacity of optical switches than copying data in switches than copying data in electronic domain.electronic domain.

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Multicast-Capable Multicast-Capable WDM NetworksWDM Networks A A connectionconnection or a or a lightpathlightpath is an ordis an ord

ered pair of nodes (ered pair of nodes (xx,,yy) correspondin) corresponding to transmission of a packet from sog to transmission of a packet from source urce xx to destination to destination yy..

The connections established betweeThe connections established between one source node and multiple destn one source node and multiple destination nodes (ination nodes (xx,,yy11), (), (xx,,yy22), …, (), …, (xx,,yynn) ar) are referred to as a e referred to as a multicast connectimulticast connectionon..

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Multicast-Capable Multicast-Capable NodesNodes In a multicast connection, a node In a multicast connection, a node

may be required to may be required to connect an connect an incoming channel to a set of incoming channel to a set of outgoing channelsoutgoing channels where each where each outgoing channel is on a different outgoing channel is on a different fiber. fiber.

In order to support multicast In order to support multicast efficiently, the routing nodes have to efficiently, the routing nodes have to have have the capability of splittingthe capability of splitting and/or and/or broadcasting from input to output.broadcasting from input to output.

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Three different multicast Three different multicast modelmodel

(a) Multicast with same (a) Multicast with same wavelength ( wavelength (MSWMSW))

(b) Multicast with same (b) Multicast with same destination destination wavelength wavelength (MSDW)(MSDW)

(c) Multicast with any (c) Multicast with any wavelength wavelength (MAW)(MAW)

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Multicast AssignmentMulticast Assignment

A A multicast assignmentmulticast assignment is a mapping from is a mapping from a set of source nodes to a a set of source nodes to a maximummaximum set set of destination nodes with of destination nodes with no overlappingno overlapping allowed among the destination nodes of allowed among the destination nodes of different source nodes.different source nodes.

An arbitrary multicast communication An arbitrary multicast communication pattern can be decomposed into several pattern can be decomposed into several multicast multicast assignments.assignments.

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Examples of multicast assignments in a 4-node network

There are a total of There are a total of NN connections in any connections in any multicast assignment. multicast assignment.

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FAN-OUTFAN-OUT

In a multicast assignment, In a multicast assignment, the the number of destination nodesnumber of destination nodes from from the same source node is referred to the same source node is referred to as the as the fan-out fan-out of the source node.of the source node.

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NonblockingNonblocking

Strictly Nonblocking (SNB):Strictly Nonblocking (SNB):For any legitimate connection request, it is For any legitimate connection request, it is always possible to provide a connection paalways possible to provide a connection path without disturbing existing connections. th without disturbing existing connections.

Wide-sense Nonblocking (WSNB):Wide-sense Nonblocking (WSNB): If the path selection must If the path selection must follow a routing afollow a routing algorithmlgorithm to maintain the nonblocking conn to maintain the nonblocking connecting capacity.ecting capacity.

Rearrangeable Nonblocking (RNB)Rearrangeable Nonblocking (RNB)

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Focus of this paperFocus of this paper

To determine To determine bounds on the minimum bounds on the minimum number of wavelengthsnumber of wavelengths (denoted as (denoted as wwrr) ) required for a multicast-capable WDM required for a multicast-capable WDM network to be rearrangeable for any mnetwork to be rearrangeable for any multicast assignments.ulticast assignments.

In other words, to determine In other words, to determine the conditthe conditionion under which any multicast assignm under which any multicast assignment can be embedded in a WDM networent can be embedded in a WDM network k offlineoffline..

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AssumptionAssumption

Each link in the network is Each link in the network is bidirectionalbidirectional..

Adopt the Adopt the MSWMSW model but model but light light splitterssplitters are available at every are available at every routing node.routing node.(Wavelength-continuity constraint)(Wavelength-continuity constraint)

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OutlineOutline

IntroductionIntroduction Multicast-Capable WDM NetworksMulticast-Capable WDM Networks Generalization of the conflict graphGeneralization of the conflict graph RingsRings MeshesMeshes HypercubesHypercubes Linear ArraysLinear Arrays Comparison and ConclusionsComparison and Conclusions

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Conflict graphConflict graph

Given a collection of connectionsGiven a collection of connections G=(V,E) : an undirected graph, G=(V,E) : an undirected graph,

wherewhereV={V={vv: : vv is a connection in the is a connection in the network}network}E={E={abab: : aa and and bb share a physical fiber share a physical fiber link}link} ( (aa, , bb can’t use the same can’t use the same wavelength.)wavelength.)

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66Conflict graph=K6

Example of Conflict Example of Conflict graphgraph

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In Multicast-capable In Multicast-capable networknetwork Those connections that belong to the

same source node do not conflict with each other. In fact, they must utilize the same wavelength under MSW model. We can treat these connections as a single connection that goes through all the links used by any connection among these connections.

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ContractionContraction

A contraction G : ab of a graph G. (a) The original graph. (b) The contracted graph.

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Modified Conflict graphModified Conflict graph

For any source node vk that is multicasting in the corresponding conflict graph G, contract all of the vertices ak1

, ak2, …, a

kn that correspond to the multicast conn

ection from source node vk into a single vertex bk.

We call the resulting graph a modified conflict graph G’.

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ExampleExample

G G’

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Modified Conflict graphModified Conflict graph

(Equivalent definition)G=(V,E) : an undirected graph, whereG=(V,E) : an undirected graph, whereV={V={vv: : vv is a is a multicast connectionmulticast connection in the in the network} network}(A source node corresponds to a vertex.)(A source node corresponds to a vertex.)E={E={abab: : aa and and bb share a physical fiber link} share a physical fiber link}((aa, , bb can’t use the same wavelength.)can’t use the same wavelength.)

Wavelength assignment Wavelength assignment Graph vertex coloring Graph vertex coloring

2222

OutlineOutline

IntroductionIntroduction Multicast-Capable WDM NetworksMulticast-Capable WDM Networks Generalization of the conflict graphGeneralization of the conflict graph RingsRings MeshesMeshes HypercubesHypercubes Linear ArraysLinear Arrays Comparison and ConclusionsComparison and Conclusions

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Rings - Theorem 1Rings - Theorem 1

The necessary and sufficient condition for a bidirectional WDM ring with N nodes to be rearrangeable for any multicast assignment under shortest path routing is the number of wavelengths wwrr = = NN /2/2..

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Meshes – Lemma 1Meshes – Lemma 1

(Under row-major routing) In a pq mesh, the degree of a vertex v in the modified conflict graph G’ associated with a source node that has a fan-out of m is (q1)+m(p2).

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Proof of Lemma 1Proof of Lemma 1

S: sourcer: the row of SDj (1j m): destinationscj: the column of Dj

In row r, at most q1 other source nodes.

In column cj, at most p-2destination nodes otherthan Dj.deg(v)(q-1)+m(p2).

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Meshes – Theorem 2Meshes – Theorem 2

The number of wavelengths required for a pq mesh to be rearrangeable for any multicast assignment under row-major routing is bounded by the following equation:

Proved by Qiao & Mei

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Lemma GLemma G

Let G be a simple graph. If, for some integer k, the number of vertices with degree k is no more than k, then G is k colorable.

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Proof of Theorem 2Proof of Theorem 2

Assume that Assume that at most at most kk vertices in G vertices in G’ with degree ’ with degree kk, where 1 , where 1 kk pqpq..(We want to find the smallest (We want to find the smallest kk..))

Then, for some Then, for some mm 0, 0,qq1+(1+(mm1)1)((pp2) < 2) < kk qq1+1+mm((pp2).2).

By Lemma 1, if deg(By Lemma 1, if deg(vv) ) k k for some vefor some vertex rtex vv, the fan-out of , the fan-out of vv mm..

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Proof of Theorem 2 Proof of Theorem 2 (contd.)(contd.) There are at most There are at most pqpq destinations destinations

kmkm pqpq ((qq1+(1+(mm1)1)((pp2))2))mm < < pqpq ((pp2) 2) mm2 2 + (+ (qqpp1) 1) mmpq pq < 0< 0

)2(2

22104 222

p

qppqqppqqpm

12

22104

)2(1

222

qppqqppqqp

mpqk

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Proof of Theorem 2 Proof of Theorem 2 (contd.)(contd.) By Lemma G,By Lemma G,

If If pp==qq==nn, the upper bound given , the upper bound given here is here is

12

22104 222

qppqqppqqp

wr

21 nnn

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OutlineOutline

IntroductionIntroduction Multicast-Capable WDM NetworksMulticast-Capable WDM Networks Generalization of the conflict graphGeneralization of the conflict graph RingsRings MeshesMeshes HypercubesHypercubes Linear ArraysLinear Arrays Comparison and ConclusionsComparison and Conclusions

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e-cube routinge-cube routing

Source Source ss destination destination dd ss = = ssnn11ssnn22…s…s11ss00 ssnn11ssnn22…s…s11dd00 ssnn11ssnn22……dd11dd00 …… ssnn11ddnn22…d…d11dd00 ddnn11ddnn22…d…d11dd00 == dd

where where

ssnn11ssnn22…s…si+i+11ssiiddnn11……dd11dd0 0 ssnn11ssnn22…s…si+i+11ddiiddnn11……dd11dd00

is called an is called an ((ii+1)th dimensional link+1)th dimensional link..

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Lemma 2Lemma 2

In an In an nn-dimensional hypercube -dimensional hypercube under e-cube routing, the number under e-cube routing, the number of of distinct source nodesdistinct source nodes that that goes goes through some (through some (ii+1)th dimensional +1)th dimensional linklink is is 22ii, and the number of , and the number of distinct destination nodesdistinct destination nodes that that goes through some (goes through some (ii+1)th +1)th dimensional link is dimensional link is 22n-1-in-1-i..

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Proof of Lemma 2Proof of Lemma 2

ssnn11ssnn22…s…si+i+11ssiiddnn11…d…d11dd0 0

ssnn11ssnn22…s…si+i+11ddiiddnn11…d…d11dd00

22n-1-in-1-i 2 2ii

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Lemma 3Lemma 3

In an In an nn-dimensional hypercube und-dimensional hypercube under e-cube routing, the degree of a ver e-cube routing, the degree of a vertex ertex vv in the modified conflict grap in the modified conflict graph h GG’ associated with a source nod’ associated with a source node that has a fan-out of e that has a fan-out of mm is less than is less than or equal toor equal to

22n

m

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Proof of Lemma 3Proof of Lemma 3

Suppose a connection Suppose a connection cc: : ssdd in a multica in a multicast connection associated with vertex st connection associated with vertex vv go goes through es through uu links: links: kk11+1, +1, kk22+1, …, +1, …, kkuu+1-th d+1-th dimensional links, where imensional links, where kk11< < kk22< …< < …< kkuu..

Let Let NNjj ={ ={wwG’G’: the multicast connection o: the multicast connection of f ww share the ( share the (kkjj+1+1))th link with th link with cc}, 1 }, 1 jj uu..

By Lemma 2, By Lemma 2,

)2,2min(|| 1 jj knkjN

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Proof of Lemma 3 (contd.)Proof of Lemma 3 (contd.)

Let Let ii be the integer such that be the integer such that when when jj ii, and, and when when jj > > ii..

That is, That is, kkjj ( (nn1)/2 when 1)/2 when jj ii, and, and kkjj > ( > (nn1)/2 when 1)/2 when jj > > ii..

jj knk 122jj knk 122

ikii NNNN 2|||| 21

11121 2||||

ikniuii NNNN

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Proof of Lemma 3 (contd.)Proof of Lemma 3 (contd.)

Since Since kkii ( (nn1)/2 =1)/2 =nn/2/21 1 and and kki+i+11 > ( > (nn1)/2 1)/2 nn11 kki+i+11 < ( < (nn1)/2 =1)/2 =nn/2/211

Therefore Therefore

Each connection in the multicast connectioEach connection in the multicast connection represented by n represented by vv contributes to the de contributes to the degree of gree of vv. Hence …. Hence …

221 2||

n

uNNN

22n

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Theorem 3Theorem 3

The number of wavelengths required for an n-dimensional hypercube to be rearrangeable for any multicast assignment under e-cube routing is bounded by

2/222 22

nn

r

n

w

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Proof of Theorem 3Proof of Theorem 3

Assume that Assume that at most at most kk vertices in G vertices in G’ with degree ’ with degree kk..(We want to find the smallest (We want to find the smallest kk..))

Then, for some Then, for some mm 0, 0,

By Lemma 3, if deg(By Lemma 3, if deg(vv) ) k k for some vefor some vertex rtex vv, the fan-out of , the fan-out of vv mm..

22 22)1(nn

mkm

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Proof of Theorem 3 (contd.)Proof of Theorem 3 (contd.)

There are at most 2There are at most 2nn connections connections

Therefore Therefore

2/2

2

2

2

2)1(

22)1(

2

n

n

nn

n

m

mm

mm

km

2/222 22

nnn

mk

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OutlineOutline

IntroductionIntroduction Multicast-Capable WDM NetworksMulticast-Capable WDM Networks Generalization of the conflict graphGeneralization of the conflict graph RingsRings MeshesMeshes HypercubesHypercubes Linear ArraysLinear Arrays Comparison and ConclusionsComparison and Conclusions

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Linear arrayLinear array

There are only two possible directions for anyconnection in a linear array and the routing algorithm is unique.

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Properties of Linear Properties of Linear ArrayArray Different direction of connections do Different direction of connections do

not have to use the same not have to use the same wavelength.wavelength.

Revise the modified conflict graph G’ Revise the modified conflict graph G’ such that each such that each same-direction same-direction multicast connectionmulticast connection is represented is represented as a vertex of G’.as a vertex of G’.

We only need to We only need to consider the longest consider the longest connectionconnection among all connections in among all connections in a same-direction multicast a same-direction multicast connection.connection.

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ExampleExample

Multicast assignment

Modified Conflict Graph

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Theorem 4Theorem 4

The necessary and sufficient conditiThe necessary and sufficient condition for a linear array with on for a linear array with NN nodes to nodes to be rearrangeable for any multicast abe rearrangeable for any multicast assignment is ssignment is the number of wavelenthe number of wavelengthsgths wwr r ==NN /2/2..

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Proof of Theorem 4Proof of Theorem 4

(Necessity: (Necessity: wwr r NN /2/2 ) )

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Proof of Theorem 4 (contd.)Proof of Theorem 4 (contd.)

(Sufficiency: (Sufficiency: wwr r NN /2/2 ) )Consider the Consider the rightwardrightward longestlongest multi multicast connections.cast connections.

Let (Let (ii11, , ii22), (), (ii22, , ii33), …, (), …, (iikk-2-2, , iikk-1-1), (), (iikk-1-1, , iikk) be ) be a maximal collection of multicast coa maximal collection of multicast connections, called nnections, called same-wavelength csame-wavelength collection (SWC)ollection (SWC), denoted as (, denoted as (ii11,,iikk). ).

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Proof of Theorem 4 (contd.)Proof of Theorem 4 (contd.)

(Sufficiency: (Sufficiency: wwr r NN/2/2 ) ) (contd.)(contd.) Each node Each node aa of the linear array can b of the linear array can belong to only one elong to only one SWC SWC (i.e., the SWC i(i.e., the SWC is (s (aa,,bb) or () or (cc,,aa)). )).

There are at most There are at most NN/2/2 SWCs SWCs NN/2/2 wavelengths are sufficient. wavelengths are sufficient.

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SummarySummary

(q-1)

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ConclusionConclusion

By introducing multicast-capable By introducing multicast-capable routing nodes, the number of routing nodes, the number of wavelengths required for wavelengths required for embedding an arbitrary multicast embedding an arbitrary multicast assignment is reduced a lot.assignment is reduced a lot.

Future work:Future work:Deriving Deriving tightertighter bounds for bounds for nn-dim -dim hypercube and phypercube and pq meshq mesh