1 wide-sense nonblocking multicast in a class of regular optical networks from: c. zhou and y. yang,...

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Wide-Sense Wide-Sense Nonblocking Multicast Nonblocking Multicast in a Class of Regular in a Class of Regular Optical NetworksOptical Networks

From:From: C. Zhou and Y. Yang, IEEE Transactions C. Zhou and Y. Yang, IEEE Transactions on on communications, vol. 50, No. 1, Jan. communications, vol. 50, No. 1, Jan. 2002 2002

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AbstractAbstract

Study Study multicastmulticast communication in communication in a class of optical a class of optical WDMWDM networks networks with regular topologies.with regular topologies.

Derive the necessary and sufficient Derive the necessary and sufficient conditions on the minimum number conditions on the minimum number of wavelengthsof wavelengths required for a WDM required for a WDM network to be network to be wide-sense wide-sense nonblockingnonblocking for multicast for multicast communication under some communication under some commonly used routing algorithm.commonly used routing algorithm.

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OutlineOutline

IntroductionIntroduction Linear ArraysLinear Arrays Rings (unidirectional, Rings (unidirectional,

bidirectional)bidirectional) Meshes and ToriMeshes and Tori HypercubesHypercubes ConclusionsConclusions

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DefinitionsDefinitions

Wavelength-division multiplexing (WWavelength-division multiplexing (WDM):DM): multiple wavelength channels f multiple wavelength channels from different end-users can be multirom different end-users can be multiplexed on the same fiber.plexed on the same fiber.

A A connectionconnection or a or a lightpath lightpath is an ordis an ordered 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..

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Multicast communicationMulticast communication: transmitting : transmitting information from a single source node to information from a single source node to multiple destination nodes.multiple destination nodes.

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

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

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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 the To determine the minimum number minimum number of wavelengthsof wavelengths required for a WDM required for a WDM network to be wide-sense network to be wide-sense nonblocking for arbitrary multicast nonblocking for arbitrary multicast assignments.assignments.

In other words, to determine In other words, to determine the the conditioncondition on which any multicast on which any multicast assignment can be embedded in a assignment can be embedded in a WDM network on-line under the WDM network on-line under the routing algorithm.routing algorithm.

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AssumptionAssumption

Each link in the network is Each link in the network is bidirectional.bidirectional.

No No wavelength converterwavelength converter facility is facility is available in the network. Thus, a available in the network. Thus, a connection must use the same connection must use the same wavelength throughout its path.wavelength throughout its path.

No No light splitterslight splitters are equipped at are equipped at each routing nodes.each routing nodes.

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OutlineOutline



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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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Theorem 1Theorem 1

The necessary and sufficient conditiThe necessary and sufficient condition for a WDM linear array with on for a WDM linear array with NN nod nodes to be wide-sense nonblocking for es to be wide-sense nonblocking for any multicast assignment is any multicast assignment is the numthe number of wavelengthsber of wavelengths www w ==N N －－ 1.1.

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Proof of Theorem 1Proof of Theorem 1

Sufficiency (Sufficiency (wwww N N －－ 1): 1): – The connections on the same link in different The connections on the same link in different

directions may use the same wavelength.directions may use the same wavelength.– There are at most There are at most N N －－ 1 connections in the sa1 connections in the sa

me direction in a multicast assignment.me direction in a multicast assignment.

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Necessity (Necessity (wwww N N －－ 1): 1):

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Wavelength assignment Wavelength assignment algorithm in a linear arrayalgorithm in a linear array

TR: the wavelengths used for rightward connections.TL: the wavelengths used for leftward connections.TW: available wavelengths Step 1: Step 1: TL = = TR = =ΦΦ, |, |TW| = | = N N －－ 1.1. Step 2: When a new rightward (leftward) connectioStep 2: When a new rightward (leftward) connectio

n is requested, assign it a wavelength in n is requested, assign it a wavelength in TW －－ TR ( (TW

－－ TL), and add this wavelength to ), and add this wavelength to TR ( (TL).). Step 3: When an existing rightward (leftward) connStep 3: When an existing rightward (leftward) conn

ection is released, delete it from ection is released, delete it from TR ( (TL). ).

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OutlineOutline



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Unidirectional Rings Unidirectional Rings

The routing algorithm is unique.The routing algorithm is unique. Assume the directional of a ring is Assume the directional of a ring is

counter-clockwise.counter-clockwise.

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Theorem 2Theorem 2

The necessary and sufficient conditiThe necessary and sufficient condition for a unidirectional WDM ring witon for a unidirectional WDM ring with h NN nodes to be wide-sense nonbloc nodes to be wide-sense nonblocking for any multicast assignment is king for any multicast assignment is the number of wavelengths the number of wavelengths wwww = = NN..

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Conflict graphConflict graph

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

V={V={vv: : vv is a connection in the network} is a connection in the network}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.)

The The chromatic numberchromatic number (G) of G is the (G) of G is the minimum number of wavelengthsminimum number of wavelengths required required for the corresponding connections.for the corresponding connections.

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Conflict graph (contd.)Conflict graph (contd.)

Find Find (G) is a NP-complete (G) is a NP-complete problem.problem.

(G) can be efficiently determined(G) can be efficiently determinedfor conflict graphs for multicast for conflict graphs for multicast communication in most of the communication in most of the networks.networks.

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Sufficiency (Sufficiency (wwww NN ) : ) : Since there are a total of Since there are a total of NN connections connections in any multicast assignment.in any multicast assignment.

Necessity (Necessity (wwww NN ) : ) : Consider the multicast assignment:Consider the multicast assignment:NN = { = {ii ((i i 1) mod 1) mod NN : 0 : 0 ii NN1}1}The conflict graph of The conflict graph of NN is K is KNN ..Therefore Therefore wwww (K(KNN)=)=NN..

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66Conflict graph=K6

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Tu: currently used wavelengths Tn: available wavelengths

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Bidirectional RingsBidirectional Rings

There are two possible paths for a connection between any two nodes: clockwise or counter-clockwise.

The shortest path routing algorithm is adopted.

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Theorem 3Theorem 3

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

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Sufficiency (Sufficiency (wwww NN /2/2 ) :) :– In a multicast assignment, there are at

most N connections.– We can divide the N connections in a m

ulticast assignment into NN /2/2 pairs with the connections in each pair using the same wavelength.

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Proof of Theorem 3Proof of Theorem 3 Necessity (Necessity (wwww NN /2/2 ) :) :

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(0,2)

(1,3)

(2,4) (3,0)

(4,1)

Proof of Theorem 3Proof of Theorem 3 Necessity (Necessity (wwww NN /2/2, , NN is odd is odd ) :) : NN = { = {ii ((i+i+((NN1)/2) mod 1)/2) mod NN : 0 : 0 ii NN 1}1}

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OutlineOutline



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Meshes and ToriMeshes and Tori

Definition 1:Definition 1:Under the Under the row-major shortest path routirow-major shortest path routingng, for a connection request , for a connection request ((((xx00, , yy00), (), (xx11, , yy11)) in a mesh or a torus,)) in a mesh or a torus,the path is deterministically from node the path is deterministically from node ((xx00, , yy00) to node () to node (xx11, , yy11) in row ) in row xx00 along alongthe shortest path first, then to node (the shortest path first, then to node (xx11, , yy11) in column ) in column yy11 along the shortest path. along the shortest path.

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MeshesMeshes

Definition 2:Definition 2: For a connection in a mesh under For a connection in a mesh under row-major shortest path routing, if row-major shortest path routing, if the connection goes right at the the connection goes right at the first step from the source, we refer first step from the source, we refer to it as a to it as a rightward connectionrightward connection. If . If the connection goes left at the first the connection goes left at the first step, we refer to it as a step, we refer to it as a leftward leftward connectionconnection. Otherwise, we refer to . Otherwise, we refer to it as a straight connection.it as a straight connection.

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Theorem 4Theorem 4

The necessary and sufficient conditiThe necessary and sufficient condition for a WDM mesh with on for a WDM mesh with pp rows and rows and q q columns to be wide-sense nonblocki columns to be wide-sense nonblocking for any multicast assignment undng for any multicast assignment under row-major shortest path routing is er row-major shortest path routing is the number of wavelengths the number of wavelengths www w = = pp((q q 1)1)

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Sufficiency (Sufficiency (wwww pp((q q 1)) :1)) :– Wavelengths: Wavelengths: ww00, , ww11, …, , …, wwpp((q q 1) 1) 11

– Let RLet Rii ={ ={wwii((q q 1)1), , wwii((qq 1)+11)+1, …, , …, ww((i+1i+1))((q q 1)1)11},},0 0 ii pp －－ 1 . (|R1 . (|Rii|= |= q q 1)1)

– Let the connections destined to row Let the connections destined to row ii use use the wavelengths within range Rthe wavelengths within range Rii..

– The connections destined to the same colThe connections destined to the same column will use different wavelengths.umn will use different wavelengths.

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Sufficiency (Sufficiency (wwww pp((q q 1)) (continued…)1)) (continued…)– Among all the connections to the same row, Among all the connections to the same row,

if the sources of two connections are in diffeif the sources of two connections are in different rows, they can use the same wavelengtrent rows, they can use the same wavelength. h. (by row-major routing)(by row-major routing)

– Consider those connections originated from Consider those connections originated from the same row and destined to the same row. the same row and destined to the same row. Since each row can be considered as a linear Since each row can be considered as a linear array with array with qq nodes. By Thm 1, nodes. By Thm 1, qq1 wavelengt1 wavelengths are sufficient for WSNB. Hence hs are sufficient for WSNB. Hence wwww pp((q q 1).1).

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Necessity (Necessity (wwww pp((qq 1)1) ) :) :

Node (0,0) is the source of rightward pp((q q 1)1) connections. They must share the link (0,0) (0,1).So pp((q q 1)1) wavelengths are required.

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TRi: currently used

wavelengths for rightward connections to row i.

TLi : currently used

wavelengths for leftward connections to row i.

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TorusTorus

Definition:Definition:A A torus networktorus network is a mesh with wrap- is a mesh with wrap-around connections in both the around connections in both the xx an and d yy directions. This allowed the most directions. This allowed the most distant processors to communicate idistant processors to communicate in 2 hops.n 2 hops.

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Theorem 5Theorem 5

The necessary and sufficient conditiThe necessary and sufficient condition for a WDM torus with on for a WDM torus with pp rows and rows and q q columns to be wide-sense nonblocki columns to be wide-sense nonblocking for any multicast assignment undng for any multicast assignment under row-major shortest path routing is er row-major shortest path routing is the number of wavelengths the number of wavelengths www w = = ppqq/2/2..

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Sufficiency (Sufficiency (wwww ppqq/2/2) :) :– (Similar to meshes) Divide the wavelength(Similar to meshes) Divide the wavelength

s to sets Rs to sets R00~R~Rpp11, and let the connections d, and let the connections destined to row estined to row ii use the wavelengths withi use the wavelengths within set Rn set Rii..

– We need only to consider those connectioWe need only to consider those connections originated from the same row and destins originated from the same row and destined to the same row. Since each row can ned to the same row. Since each row can be considered as a bidirectional ring with be considered as a bidirectional ring with qq nodes. By Thm 3, nodes. By Thm 3, qq/2/2 wavelengths are s wavelengths are sufficient for WSNB. Hence ufficient for WSNB. Hence wwww ppqq/2/2..

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Necessity (Necessity (wwww ppqq/2/2) when ) when qq is even: is even: Consider the connection (0,0) Consider the connection (0,0) ( (ii, , jj),), where 1 where 1 ii pp1, 0 1, 0 jj qq-1.-1.

q=6

There areThere are pp ( (qq/ / 2)2) connections passed through the link(0,0)(0,1) or (0,0)(0, q-1).

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Necessity (Necessity (wwww ppqq/2/2) when ) when qq is odd: is odd: Consider the connection Consider the connection (0, (0, jj) ) ( (ii, , jj+(+(qq1)/21)/2 modmod qq),), where 0 where 0 i i pp1, 0 1, 0 jj qq-1.-1.

q=5

The connectionThe connection(0, (0, jj) ) ( (ii, , jj+(+(qq1)/21)/2 modmod qq))must pass through the node must pass through the node (0, (0, jj+(+(qq1)/21)/2 modmod qq)). So there . So there are are pp connections pass connections passthrough the linkthrough the link (0,(0, jj) ) (0, (0, jj+(+(qq1)/21)/2 modmod qq))..By Thm 3, a ring with By Thm 3, a ring with qq nodes nodes

need (need (qq1)/2+1 wavelengths.1)/2+1 wavelengths.Hence Hence ppqq/2/2 wavelengths is wavelengths isnecessary.necessary.

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( )

( )

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OutlineOutline



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HypercubesHypercubes

Definition: (e-cube routing)Definition: (e-cube routing)In an In an nn-cube with -cube with NN=2=2nn nodes, let eac nodes, let each node h node bb be binary-coded as be binary-coded as bb = = bbnnbbnn11…b…b22bb11, where the , where the iist bit corresst bit corresponds to the ponds to the iith dimension.th dimension.In e-cube routing, a route from node In e-cube routing, a route from node ss = = ssnnssnn11…s…s22ss11 to node to node dd = = ddnnddnn11…d…d22dd1 1 is uis u

niquely determined as follows:niquely determined as follows:

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HypercubesHypercubes

Definition: (e-cube routing) (contd.)Definition: (e-cube routing) (contd.) ss = = ssnnssnn11…s…s22ss11 ssnnssnn11…s…s22dd11 ssnnssnn11…d…d22dd11 …… ssnnddnn11…d…d22dd11 ddnnddnn11…d…d22dd1 1 == dd

Note that if Note that if ssii==ddii, no routing is needed al, no routing is needed along dimensionong dimension i i . .

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Example of e-cube Example of e-cube routingrouting

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A hypercube can be divided into two sA hypercube can be divided into two subcubes:ubcubes:0-subcube={0-subcube={bbnnbbnn11…b…b2200: : bbi i {0,1}}{0,1}}1-subcube={1-subcube={bbnnbbnn11…b…b2211: : bbi i {0,1}}{0,1}}

For a connection in a hypercube, if the For a connection in a hypercube, if the destination of this connection is in the destination of this connection is in the 0-subcube0-subcube ( (1-subcube1-subcube), it is referred t), it is referred to as a o as a 0-connection0-connection ( (1-connection1-connection).).

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Theorem 6Theorem 6

The necessary and sufficient conditiThe necessary and sufficient condition for a WDM hypercube with on for a WDM hypercube with NN=2=2n n nnodes to be wide-sense nonblocking fodes to be wide-sense nonblocking for any multicast assignment under or any multicast assignment under e-cube routing is the number of wave-cube routing is the number of wavelengths elengths www w = = NN//2 = 22 = 2nn11..

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Sufficiency (Sufficiency (wwww NN//2 ) :2 ) :– At most At most NN/2 connections go from the /2 connections go from the

1(or 0)-subcube to the 0(or 1)-subcube.1(or 0)-subcube to the 0(or 1)-subcube.(The first step must go to the other subcu(The first step must go to the other subcube by e-cube routing.)be by e-cube routing.)

– At most At most NN//2 0/1-connections are in the 2 0/1-connections are in the 0/1-subcube.0/1-subcube.

– Any 0-connection and any 1-connection cAny 0-connection and any 1-connection cannot interfere with each other since theannot interfere with each other since they are in different directions or in different y are in different directions or in different subcubes.subcubes.

NN//2 wavelengths are sufficient.2 wavelengths are sufficient.

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Proof of Theorem 6Proof of Theorem 6 Necessity (Necessity (wwww NN/2): Consider the case /2): Consider the case { 00…0 { 00…0 bbnnbbnn11……bb2211 and and 11…1 11…1 bbnnbbn-n-11…b…b2200 : : bbii{0,1}{0,1} }}

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T0 : used wavelengths for 0-connectionsT1 : used wavelengths for 1-connections Tw : available wavelengths

-1

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ConclusionConclusion

Summary:

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ConclusionConclusion

Future work:Future work:– Generalize the approach developed in tGeneralize the approach developed in t

his paper to his paper to other routing algorithmsother routing algorithms an and d other network topologiesother network topologies including irr including irregular networks.egular networks.

– Determine nonblocking conditions for Determine nonblocking conditions for multicast assignments in WDM networkmulticast assignments in WDM networks with s with wavelength converterswavelength converters and/or and/or liglight splitting switchesht splitting switches..

1 wide-sense nonblocking multicast in a class of regular optical networks from: c. zhou and y. yang,...

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