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A new scalable optimal topology for multi-hop optical networks

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A new scalable optimal topology for multi-hop optical networks
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  A new scalable optimal topology for multi-hop optical networks U. Bhattacharya a, *, R. Chaki b a  Department of Computer Science and Technology, Bengal Engineering College D.U., P.O.B. Garden, Howrah 711 103, India b  Joint Plant committee, 52/1A, B. C. Road, Calcutta 700 019, India Received 8 December 2003; revised 3 September 2004; accepted 14 September 2004Available online 7 October 2004 Abstract This paper presents a scalable optimal logical topology scale-net for multi-hop optical networks based on de Bruijn graph, a regular non-scalable one with simple routing strategy. In addition to its scalability property, this new topology maintains the simplicity in routing as in deBruijn graph while keeping its diameter same throughout the insertion of nodes as that in a de Bruijn graph where the diameter is of logarithmic value of its number of nodes. Also, perturbation in the network is maintained at a very low level, while inserting a node in thenetwork. Simulation results also show a reasonable average hop distance between any source destination pair of the topology developed. q 2004 Elsevier B.V. All rights reserved. Keywords:  de Bruijn graph; Multi-hop network; Routing; Scalable; Diameter; Optimal 1. Introduction WDM [15] is the approach used in high-speed widebandfibers in optical networks [13,17,18] for propagating severalchannels at different wavelengths on a single fiber,exploiting the huge bandwidth of the optical fibers totransmit information between any source-destination pair.End-users in a WDM-based backbone network maycommunicate with one-another via all-optical channels,which are referred to as light-paths.A light-path [15] between end-nodes is a path betweenthem through router(s) or star coupler(s) using a particularwavelength (often called a channel) for each segment of thepath traversed. In an  N  -node network, if each node isequipped with  N  K 1 transceivers and if there are enoughwavelengths on all fiber links, then every node-pair could beconnected by an all-optical light-path. But the cost of thetransceivers dictate us to equip each node with only a few of them resulting to limit the number of WDM channels in afiber to a small value  d  . Thus, only a limited number of light-paths may be set up on the network.Typically, the physical topology of a network consists of nodes and links in a broadcast star or ring or bus. On anyunderlying physical topology, one can impose a carefullyselected connectivity pattern that provides dedicated con-nections between certain pair of nodes. Traffic destined to anodethatisnotdirectlyreceivingfromthetransmittingnode,must be routed through the intermediate nodes. The overlaidtopology is referred to as multi-hop logical topology [17].GEM-net [1], de Bruijn graph [11], shuffle-net [6], etc. are examples of such existing multi-hop logical topologies.Since two different light-paths cannot be assigned thesame WDM channel if they share the same fiber [8], thedemand for the number of channels per fiber will be higherif many light-paths have to share the same fiber. As thelength of the source–destination path is bounded by thediameter (distance between two nodes of the network thatare furthest from each other), the requirement for thenumber of channels per fiber will be small for a low-diameter topology. In a multi-hop environment, thecommunication delay may be proportional to the numberof intermediate routers the packets have to pass through ontheir journey from the source to the destination and as aconsequence, in such an environment a topology with asmall diameter is quite attractive.The logical topology [14] is usually represented by adirected graph (or digraph)  G Z ( V  ,  E  ) (where  V   is the set of  0140-3664/$ - see front matter q 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.comcom.2004.09.003Computer Communications 28 (2005) 557–570www.elsevier.com/locate/comcom* Corresponding author. Tel.:  C 91 33 2668 5154; fax:  C 91 33 26682916. E-mail addresses:  uma_bh2000@yahoo.co.in (U. Bhattacharya),ub@cs.becs.ac.in (U. Bhattacharya).  nodes and  E   is the set of edges) with each node of   G representing a node of the physical topology and each edge(denoted by  u / v ) representing a direct all optical light-pathfrom node u to node  v . The graph is desired to have smallnodal degree for  low cost  ,  simple routing  scheme, smalldiameter for  high-speed communication  and  growth capa-bility  with least possible perturbation in the network tosatisfy the optimality criteria. A regular topology (having astructured node-connectivity pattern) has simpler routingschemes and can support a large number of nodes with asmall diameter and small nodal degree. But scalability[12,19] remains a problem with such regular structures(where number of nodes is some discrete integer value) in aLAN/MAN/WAN environment, where number of nodesundergo frequent changes. Irregular multi-hop structureshave the advantage of addressing the optimality criteriondirectly [14] at the cost of lack of structural connectivitypattern which makes the routing more complex.The topology to be described in this paper seeks to designa scalable [12,19] logical irregular multi-hop topologymaintaining the simple routing scheme, small nodal degree,and low constant diameter for high-speed communication asin a regular multi-hop structure. In this work, de Bruijngraph is considered as an ideal regular base structure of theirregular multi-hop topology to be designed.In Section 2, review of the existing works on logicaltopologies and scope of the work has been mentioned.Problem to be discussed in this paper forms the content of Section 3 following which Section 4 deals with theinterconnection pattern of the new topology  scale-net   forany number of nodes and also the change in theinterconnection pattern as soon as a node is inserted inthe network. Section 5 deals with salient features of thetopology such as degree of any node, diameter of thetopology, perturbation in the network during its growth androuting scheme. Simulation results have been mentioned inSection 6 showing a reasonable number of average hopdistances between any node pair in the network. Section 7deals with the conclusion. 2. Review and scope of the work An adequate evaluation of the work can only beunderstood in the backdrop of the existing works in therelated fields. A brief overview of the same is presentedhere. The main points of interest with which this work isrelated are diameter, scalability and simple routing schemeassociated with logical topologies considered so far. 2.1. Multi-mesh (MM) and incomplete multi-mesh (IMM)network  MM architecture being a regular topology and srcinallyproposed for parallel processing systems [2–5] is definedfor  n 4 nodes for any integer  n . The diameter [2] of a complete MM network with  N   nodes where  N  Z n 4 nodes is 2 n Z 2  N  1/4 . A two-dimensional torus (comparablewith MM) with exactly same number of nodes and edgeswill have a diameter  N  1/2 [10]. So, MM network is quiteattractive in this perspective when compared with a two-dimensional torus network. It may be noted that althoughthe authors in [9] refer to the network discussed in theirpaper as the MM network, in the research community thisnetwork is known as the Manhattan Street Network  [7]. Theinterconnection pattern in MM topology proposed in [2–5]is an extension of the simple mesh connection. In an  n ! n mesh, the processors are arranged in  n  rows and  n  columns.The MM uses this as a building block for the construction of the network. The idea is to use  n 2 such blocks arrangedagain in  n  rows and  n  columns. Thus an MM network hasexactly  n 4 nodes. Each of these  n 4 nodes are identified with afour tuple label ( a , b ,  x ,  y ). The first two ( a , b ) identify theblock and the last two (  x ,  y ) identify the node within a block.Each of these co-ordinates can take a value between 1 and  n (both inclusive). The  n ! n  nodes within each block areconnected as a regular two-dimensional mesh. The inter-block connections are made using the following rules:(1)  c b , 1 % b % n , the node ( a ,  b , 1,  y ) is connected to thenode (  y ,  b ,  n ,  a ) where 1 %  y ,  a % n .(2)  c a , 1 % a % n , the node ( a ,  b ,  x , 1) is connected to thenode ( a ,  x ,  b ,  n ) where 1 %  x ,  b % n .It may be observed that the interconnection rules givenabove generates a regular topology where each node isconnected to exactly four edges. A partially completed MMnetwork with 81 nodes is shown in Fig. 1(a).The topology IMM defined for any number of nodes  N  ,1 %  N  % n 4 for some integer  n  proposed in [10] is ageneralization over MM. This topology has  m Z d  N   /  n 2 e blocks where the blocks will be arranged in rows andcolumns as in MM. The last row of blocks out of   d m  /  n e  rowsmay be partially complete with mod( m , n ) blocks while otherrows have  n  number of complete blocks. A complete block will have  n ! n Z n 2 nodes. A partially complete block with q  nodes will have  b q  /  n c  complete rows with each row havingexactly  n  nodes. The last row within a block may bepartially complete with mod( q , n ) nodes. An IMM with 40nodes is shown in Fig. 1(b).The topology IMM is incrementally expandable, i.e. an  N  -node network can be extended to  N  C 1 node network without major changes in the existing structure. Thisincremental expansion possibility of IMM makes it veryattractive for use in a LAN/MAN/WAN [16,17] environ-ment. Advantage of IMM is also that it allows theconstruction of a network for any number of nodes. Interms of simplicity of interconnection and routing thearchitecture is comparable to the regular mesh and torus.But, diameter of an IMM network with  N   nodes (1 %  N  % n 4 )and blocks of size  n ! n  increases at most to 4 n  as soon asthe network grows. U. Bhattacharya, R. Chaki / Computer Communications 28 (2005) 557–570 558  2.2. de Bruijn graph A  d  k  de Bruijn graph [11] having  N  ( Z d  k  ) nodes is anelegant regular topology with degree  d   and diameter  k  .Fig. 1(c) represents a eight node de Bruijn graph withdiameter 3 and degree 2.Each node has indegree and outdegree  d   and the diameterof the graph is  k   with the set of nodes {0,1,2, . , d  K 1} k  withan edge from node  a 1 a 2 . a k   to node  b 1 b 2 . b k   iff thecondition  b i Z a i C 1  is satisfied where  a i , b i  belongs to  A ,  A Z {0,1,2, . , d  K 1}, 1 % i % k  K 1.There is a one-to-one correspondence between allpossible states of the  d  -ary shift register of length  k   andthe nodes of de Bruijn graph  G ( d  , k  ). There is an edge joiningnode  x i  to node  x  j  if node (state)  x  j  can be reached from state  x i  with one shift (and a new input digit). In other words, deBruijn graph is the transition diagram of the shift register.From the shift-register analogy, a node (or state) in deBruijn graph can be represented by a string (or sequence) of  k   digits. An edge from node  A  to node  B  can be representedby a string of   k  C 1 digits, the first  k   digits representing node  A  and the last  k   digits representing node  B . Similarly, anypath length  l  hops can be represented by a string of   k  C 1digits. In any graph with maximum out-degree  d  , there canbe at most  d  K 1 node-disjoint paths between every pair of nodes.In determining the shortest path from node  A Z ( a 0 a 1 a 2 . a k  K 1 ) to node  B Z ( b 0 b 1 b 2 . b k  K 1 ), one needs toconsider the last several digits of   A  and the first severaldigits of   B  to obtain a perfect match over the largest possiblenumber of digits. If this match is of size  m  digits,( b 0 b 1 . b k  K m K 1 ) Z ( a m a m C 1 . a k  K 1 ), then the  m -hop short-est path from node  A  to node  B  is given by ( a 0 a 1 . a k  K 1 b k  K m b k  K m C 1 . b k  K 1 ).For example if we like to find the shortest path from 000to 101 the path will be as follows:000 / 001 / 010 / 101.It possesses a very simple routing scheme along withsmaller diameter (of the order of log d  N  ) compared to MMnetwork.This has also been observed [17] that for same value of average number of hops between any source–destinationpair, topologies based on de Bruijn graph can support alarger number of nodes than those based on shuffle-net, apopular regular logical topology. 2.3. Other existing topologies In [1], the principle of interconnection between nodes ina shuffle-net [6] is generalized (the generalized version canhave any number in each column) to obtain a scalableregular topology with reasonable diameter called GEMnet.A similar idea of generalizing Kautz graph has been studiedin [8] showing a better diameter and network throughputthan GEMnet. Adding new nodes to such networks (e.g.GEMnet, etc.) requires a  major change  in routing scheme.For example, in a multi-star implementation, a large numberof retuning of transceivers and/or renumbering nodes areneeded for [1].Considering the bi-directional ring network  [20] aslogical topology, it has been observed that adding a newnode to a bi-directional ring network involves redefining afixed number of edges and can be repeated infinitely with avery little perturbation in the network. Fig. 1. (a) MM network with 81 nodes (all inter-block lines are not shownfor clarity). (b) IMM network with 40 nodes. (c) de Bruijn network. U. Bhattacharya, R. Chaki / Computer Communications 28 (2005) 557–570  559  2.4. Scope of the work  Our motivation is to develop a topology, which has theadvantages of a ring network with respect to scalability andadvantages of a regular topology with respect to lowdiameter and simple routing. In other words, our topologyhas to satisfy the characteristics such as  small  and  constant  diameter,  simple  and  unchanged routing scheme  throughoutits growth and elegant  scalability  scheme so that new nodescan be added to the network   indefinitely  with the  least  possible perturbation  in the network.The new topology  scale-net   proposed in this paperpossess the similarity with IMM-net (a generalization overMM-net) that the structure has been developed byconsidering a regular structure (de Bruijn graph) as base.Advantage of scale-net over de Bruijn graph is that it allowsthe construction of a network with any number of nodes andthe network is incrementally expandable which makes itvery attractive for use in a LAN/MAN/WAN environment.In terms of simplicity, interconnection and routing thearchitecture is comparable to de Bruijn graph. In addition,routing scheme remains unaltered and simple throughout itsexpansion, diameter remains low and constant (as logarith-mic value of number of nodes). 3. Problem definition For any integer  d   and  k  , the proposed topology  scale-net  becomes a de Bruijn graph when number of nodes in thenetworkequals( d  ) k  and(2 d  ) k  ,where d  and2 d  arethedegreeof thegraphs,respectively,and k  isthediameter.Inthesituation,where  d  k  ! number of nodes ! (2 d  ) k  , the irregular graphstructure while keeping its diameter to the constant lowvalue k  (oftheorderoflog  N  ,where  N  isthenumberofnodesinthe network), still should maintain a simple routing scheme.Again, during insertion of nodes one after another, pertur-bation in the network should be maintained at a low level. 4. The proposed new topology scale-net Design of the proposed growing optimal topology  scale-net   as soon as insertion of any node goes on in the network has been described in Section 4.1. The design of   scale-net  for any number of nodes has been described in Section 4.2. 4.1. Design of scalable optimal interconnection topologyscale-net  Two cases may arise in this context depending on thenumber of nodes (  N  ) in the network. 4.1.1. Case 1: N  Z d  k  and N  Z (2d) k  The proposed interconnection topology gets the structureof a ( d  ,  k  ) de Bruijn graph ( d  R 2,  k  R 2) when the number of nodes (  N  ) in the network equals  d  k  . Each node has indegreeand outdegree  d   and the diameter of the graph is  k   with theset of nodes {0,1,2, . , d  K 1} k  with an edge from node a 1 a 2 . a k   to node  b 1 b 2 . b k   iff the condition  b i Z a i C 1  issatisfied where  a i , b i  belongs to  A ,  A Z {0,1,2, . d  K 1},1 % i % k  K 1.  Nodes present in d  k  de Bruijn network istermed as phase-0 nodes. Theproposedtopology scale-net  alsoassumesthestructureof a de Bruijn graph when the number of nodes (  N  ) in thenetworkequals(2 d  ) k  ,where2 d  istheindegreeoroutdegreeof each node and  k   is the diameter with the set of nodes{0,1,2, . ,2 d  K 1} k  , with an edge from node  a 1 a 2 . a k   to node b 1 b 2 . b k   iff the condition  b i Z a i C 1  is satisfied where  a i , b i belongs to  Z  ,  Z  Z {0,1,2, . 2 d  K 1}, 1 % i % k  K 1.We nowconsider a set  X  Z  Z  K  A Z { d  , d  C 1, . 2 d  K 1}. 4.1.2. Case 2: d  k  !  N  ! (2d) k  The proposed interconnection topology, when  d  k  !  N  ! (2 d  ) k  , assumes an insertion strategy (to be described inSection 4.1.2.2) of the sequence of nodes to be connected inthe network in  k   number of phases (to be describedin Section 4.1.2.1). 4.1.2.1. Node pattern in k phases.  The node pattern in eachphase (where subscript denotes the value and superscriptdenotes the position in its representation) are as shown inFig. 2.Each phase  i  is divided into 2 i K 1 no. of subphases. Nodepatterns in each subphase follow the sequence as shown inFig. 3.  Illustrative example.  Initially, the topology assumes thestructure of a de Bruijn graph of 2 2 number of nodes Fig. 2. A table showing the number of nodes and the node patterns followedin each phase of insertion.Fig. 3. Node patterns in each phase. U. Bhattacharya, R. Chaki / Computer Communications 28 (2005) 557–570 560  numbered as 00, 01,10 and 11, where  d  Z 2,  k  Z 2 and  A Z {0,1}. Fig. 4 shows a 2 2 de Bruijn graph.Hence  X  Z {2,3};  Z  Z {0,1,2,3}.The nodes to be inserted follow the sequence mentionedbelow (as obtained from Figs. 2 and 3)Phase-1: 02,03,12,13Phase-2: (a) 20,21,30,31(b) 22,23,32,33Fig. 5 shows the network after insertion of all nodes asmentioned in Figs. 2 and 3, where the nodes present in 2 2 deBruijn graph are shown connected by bold lines. Link connecting to 0* implies existence of multiple links to nodesnumbered 00, 01,02 and 03. The same is applicable to otherlinks to 1*, 2* and 3* as shown in Fig. 5. 4.1.2.2. Insertion strategy of nodes.  Several new terminol-ogies in this context need to be defined in Section 4.1.2.2.1following which the algorithm of insertion will be defined inSection 4.1.2.2.2 and an example illustrating the insertionstrategy will follow in Section 4.1.2.2.3. 4.1.2.2.1. Terminology.  The node to be inserted in  i thphase is of the form  a 00 a 11 a 22 . a k  K i K 1 k  K i K 1  x k  K ik  K i  z k  K i C 1 k  K i C 1 .  z k  K 1 k  K 1 ,(1 % i % k  ):(a)  Real children . Representation of real children of thenode (following de Bruijn convention) is as follows: a 01 a 12 . a k  K i K 2 k  K i K 1  x k  K i K 1 k  K i  z k  K ik  K i C 1 .  z k  K 2 k  K 1   where * in ( k  K 1)thposition of the pattern may take any value from  Z  (1 % i % k  ).(b)  Temporary children . Temporary children may bederived from the real children by subtracting  d   from  x -digitin the leftmost position of its representation. Hence,representation of temporary children of the node to beinserted in  i th phase is as follows: a 01 a 12 . a k  K i K 2 k  K i K 1 ð  x k  K i K 1 k  K i  K d  Þ  z k  K ik  K i C 1 .  z k  K 2 k  K 1  , where * in ( k  K 1)th position of the pattern may take any value from  Z  (1 % i % k  ).(c)  Real parent  . Representation of real parent of the node(following de Bruijn convention) is:  a 10 a 12 . a k   ik  K i K 1  x k  K i C 1 k  K i  z k  K i C 2 k  K i C 1 .  z k  K 1 k  K 2 , where * in 0th positionof the pattern may take any value from  Z  (1 % i % k  ).(d)  Temporary parent  . Representation of temporarychildren of the node to be inserted in  i th phase is as follows: a 01 a 12 . a k  K i K 2 k  K i K 1 ð  x k  K i K 1 k  K i  K d  Þ  z k  K ik  K i C 1 .  z k  K 2 k  K 1  , where * in( k  K 1)th position of the pattern may take any value from  Z  (1 % i % k  ).In this situation, node  a 00 a 11 a 22 . a k  K i K 1 k  K i K 1  x k  K ik  K i  z k  K i C 1 k  K i C 1 .  z k  K 1 k  K 1 ,1 % i % k   becomes he temporary parent of the node a 01 a 12 . a k  K i K 2 k  K i K 1 ð  x k  K i K 1 k  K i  K d  Þ  z k  K ik  K i C 1 .  z k  K 2 k  K 1  , where * in ( k  K 1)thpositionofthepatternmaytakeanyvaluefrom  Z  ,1 % i % k  . 4.1.2.2.2. Algorithm of insertion. 1 Nodes are selected in sequence as described by Figs. 2and 3 and the following steps 2–5 would be executed foreach insertion.2 Each node is connected with at least d number of realparents (ref. 5.1—Fig. 8). Fig. 4. A 2 2 de bruijn graph, with  d  Z 2 and  k  Z 2.Fig. 5. The network (having 4 2 nodes ) after insertion of all (2 d  ) k  K d  k  no. of nodes. U. Bhattacharya, R. Chaki / Computer Communications 28 (2005) 557–570  561
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