Scalable, and Modular Multihop Network Based. Jason Inessy, Subrata Banerjeez and Biswanath Mukherjeey. Abstract

Transcription

1 GEMNET: A Generalized, Shue-Exchange-Based, Regular, Scalable, and Modular Multihop Network Based on WDM Lightwave Technology 1 Jason Inessy, Subrata Banerjeez and Biswanath Mukherjeey ydepartment of Computer Science z Dept. of Electrical & Comp. Eng. University of California, Davis, CA University of Miami, Coral Gables, FL finess,mukherjeg@cs.ucdavis.edu Abstract banerjee@ece.miami.edu GEMNET is a generalization of shue-exchange networks and it can represent a family of network structures (including Shuenet & de Bruijn graph) for an arbitrary number of nodes. GEMNET employs a regular interconnection graph with highly desirable properties such as small nodal degree, simple routing, small diameter, and growth capability (viz. scalability). GEMNET can serve as a logical (virtual), packet-switched, multihop topology which can be employed for constructing the next generation of lightwave networks using wavelength-division multiplexing (WDM). Various properties of GEMNET are studied in this paper. Index Terms: shue exchange, scalable, multihop, lightwave network, WDM. 1 Introduction An attractive approach tointerconnect computing equipment (nodes) in a high-speed, packetswitched network is to employ a regular interconnection graph. It is highly desirable that the graph have (1) small nodal degree (for low network cost), (2) simple routing (to allow fast packet processing), (3) small diameter (for short message delays), and (4) growth capability, viz. the graph should be scalable (i.e., nodes can be added to it at all times) with a modularity of unity (i.e., it should always be possible to add one node to (or delete one node from) an existing (regular) graph while maintaining regularity [13]). We examine such a new network structure, called GEneralized shue-exchange Multihop NETwork (GEMNET). GEMNET can serve as a logical (virtual), multihop topology for constructing the next generation of lightwave networks using wavelength-division multiplexing (WDM), as explained below. Given a low-loss optical bandwidth of approximately 30 Terabits per second (Tbps) and a peak electronic processing speed of a few Gigabits per second (Gbps), innovative parallelism and concurrency mechanisms are needed to exploit this huge electro-optic bandwidth mismatch [1, 2, 5, 8, 9, 11]. WDM has emerged as the most promising choice 1 Supported in parts by ARPA Contract No. DABT63-92-C-0031 and NSF Grant No. NCR

2 Iness, Banerjee, & Mukherjee: GEMNET 2 since, unlike other alternatives, it only requires end-user equipment to operate at the bit rate of a WDM channel (peak electronic speed). However, practical considerations such as large channel spacing for reduced crosstalk limit the number of WDM channels in a ber today. Some of the key components for building a WDM optical network are broadcast passive optical star couplers, wavelength-routing switches, wavelength-agile tunable transmitters, and tunable receivers [1, 8, 11]. A local optical network can be constructed as follows. Congure the system as a broadcast-and-select network in which (1) nodes are connected to a passive star coupler via two-way bers, (2) information from the various nodes are combined in the star coupler, and (3) the mixed optical information is broadcast to all outputs. (See Fig. 1.) By tuning its transmitter(s) to one or more wavelength channels, a node can transmit into those channel(s) similarly, a node can tune its receiver(s) to receive from the appropriate WDM channels. By exploiting the transceiver tuning properties, one can construct either single-hop [8] ormultihop [9] network architectures. GEMNET is a regular multihop network architecture, it is a generalization of shue exchange networks, it represents a family of network structures, and it includes the wellstudied ShueNet [6] and de Bruijn graph [12] as members of its family. Fig. 1(b) shows a multihop (logical) network (a 10-node GEMNET) embedded on the physical star topology network of Fig. 1(a). In general, by using wavelength-routing switches, one can construct wide-area, multihop optical networks as well. (See, for example, [3, 11].) In a (K,M,P) GEMNET, K M nodes { each of degree P { are arranged in a cylinder of K columns and M nodes per column so that nodes in adjacent columns are arranged according to a generalization of the shue-exchange connectivity pattern using directed links [10]. The generalization allows any number of nodes in a column as opposed to the constraint ofp K nodes in a column. The logical topology in Fig. 1(b) is a (2,5,2) GEMNET. In GEMNET, there is no restriction on the number of nodes as opposed to the cases in ShueNet and de Bruijn graph which can support only KP K and P D nodes, respectively, where K D =1 2 3 and P = That is, GEMNET can represent arbitrarysized networks in a regular graph conversely, for any network size, at least two GEMNET congurations exist { one with K=1, and the other with M=1. GEMNET is also scalable in units of K if the nodes are equipped with either tunable transmitters or tunable receivers. After describing the GEMNET architecture, we study its construction, routing, mean as well as bounds on the hop distance, algorithms for balancing its link loads, and algorithms to add nodes to (and delete nodes from) an existing GEMNET. We conclude with a discussion on which conguration of GEMNET is best based on dierent network parameters.

3 Iness, Banerjee, & Mukherjee: GEMNET 3 2 GEMNET Architecture 2.1 Interconnection Pattern Let N denote the number of nodes in the network. IfNisevenly divisible by aninteger y, there exists a GEMNET with K = y columns. In the corresponding (K M P) GEMNET, the N = K M nodes are arranged in K columns (K 1) and M rows (M 1) with each nodehaving degree P. Nodea (a =0 1 2 N ; 1) is located at the intersection of column c (c =0 1 2 K ; 1) and row r (r =0 1 2 M ; 1), or simply location (c r), where c =(amod K) and r = ba=kc. The P links emanating out of a node are referred to as i-links, i =0 1 2 P ; 1. The i-link from node (c r) is connected to node (^c [^r + i] mod M ), for i =0 1 2 P ; 1 where ^c =[c +1] mod K and ^r =[rp ]. Note that, for a given number of nodes N, there are as many (K,M,P) GEMNETs as there are divisors for N. Specically, whenk = 1orM = 1,we can accommodate anysized network. However, M = 1 results in a ring with P parallel paths between consecutive nodes. This case is not considered further due to its large (O(N)) hop distance. Moreover a (K,M,P) GEMNET reduces to a ShueNet when M = P K also, when M = P D and K =1, it reduces to a de Bruijn graph of diameter 2 D, where D = GEMNET's diameter is obtained as follows. Starting at any node, note that each and every node in a particular column can be reached for the rst time on the dlog P M e th hop. This means that there were one or more nodes not covered in the previously-visited column. Due to the cylindrical nature of GEMNET, the nodes in this column will be nally covered in an additional K ; 1 hops. Thus, D = dlog P M e + K ; Routing Let (c s,r s )and(c d,r d ) be the source node and the destination node, respectively. We dene the \column distance" as the minimum hop distance in which the source node touches (covers) a node (not necessarily the destination) in the destination node's column. When c d c s, = c d ; c s because (c d ; c s ) forward hops from any node in column c s will cover a node in column c d. When c d <c s, is given by =(c d + K) ; c s because, after \sliding" c d forward by K (i.e., c d + K), due to wraparound, the situation becomes the same as when c d c s.thus, can be generalized as: =[(K + c d ) ; c s ] mod K. 2 The diameter of a network is dened as the longest distance it takes to get from an arbitrary node to another arbitrary node while taking the shortest possible path.

4 Iness, Banerjee, & Mukherjee: GEMNET 4 The hop distance from source node (c s r s ) to destination node (c d r d )isgiven by the smallest integer h of the form ( + jk), j =0 1 2, satisfying the following expression: R = h M + r d ; r s P h i mod M mod M < P h : (1) R, called the route code, species a shortest route from the source node to the destination node when it is expressed as a sequence of h base-p digits. For example, if R =(11) base 10 P = 3, and h = 4, then R can be represented as (0102) base 3. In general, if R =[ 1 2 h ] base P, then the node about to send the packet on its j th hop will route the packet to its th j outgoing link. The maximum number of iterations needed to solve forr is just dd=ke, where D is the diameter of the network. When K=1, the number of iterations is maximum and the complexity in computing R is O(log P N). Such a simplied routing scheme is one of the main advantages of a regular structure. To explain Eq. (1), we rst note that is the minimum number of hops required to reach a node in column c d from a node in column c s.however, multiple passes around the GEMNET may be required to reach the destination node, i.e., h will be of the form + jk, where j = Dene an all-0-link path to be the path traced, from a particular source node, by taking the 0-link out of every intermediate node (including the source node) for an arbitrary number of hops. Now, note that [r s P h ] mod M is the row index of the node in column c d reachable from the source node in h hops, by following the all-0-link path. Then, (h ; 1) 0-links followed by a 1-link leads to the node with row index [r s P h +1] mod M in column c d, and so on. However, on the h th hop, a maximum of P h nodes can be covered. The node reached on the h th hop from the source node by following the all-(p ; 1)-link path (dened similar to the all-0-link path) will be [r s P h +(P h ; 1)] mod M.Thus, if R is less than P h (which means that the destination node falls somewhere between the all-0-link path and the all-(p ; 1)-link path), then the destination node is reachable in h hops and its route code is given by R. In Eq. (1), the addition of M and the mod operations are required to accommodate the wraparound of row indices. Often, the P h nodes covered on the h th hop could be greater than the number of nodes in that column. This means that multiple shortest paths may exist to some nodes in that column. Having calculated R, if (R + xm) <P h for x =1 2 3,then(R + xm) is also a routing code with path-length h for any x that satises this inequality. Thus, if the shortest path from node a to node b is h hops, the number of shortest paths is given by Y = d(p h ; R)=M e (2) Hence, for a given N, the number of alternate shortest paths increases as M decreases. The

5 Iness, Banerjee, & Mukherjee: GEMNET 5 larger the number of shortest paths, the more opportunity there is to route a packet along a less-congested path and the greater is the network's ability torouteapacket along a minimum-length path when a link or node failure occurs. The tradeo is that decreasing M will increase K, which, in general, will cause the average hop distance to increase. 3 GEMNET Properties In this section, we examine dierent shortest-path routing algorithms to balance GEMNET's link loads, and study how dierent GEMNET congurations perform relative to one another. 3.1 Routing Algorithms for Balancing Link Loading In GEMNET, as in any other multihop network, an important goal is to balance the trac on dierent links as much as possible. If a link's utilization is heavy, the corresponding link queue could become long and the delay for packets traversing that link could become signicantly high. We examine link loading properties of GEMNET by considering a uniform load which requires one unit of trac to be moved between every source-destination pair. So far, we have assumed that only shortest-length paths between source-destination pairs are chosen for routing. However, if multiple shortest paths exist, trac can be routed over that shortest path which balances link ows as much as possible. The routing algorithm of Section 2.2 calculates R. Ifwechoose to always route with the base-r route code of Eq. (1) (henceforth called the \xed" routing scheme), the link loads tend to become unbalanced because, whenever there are multiple shortest paths, the \base-r" valueisused. Two alternative routing schemes, called \partially balanced" and\random", have been examined, and are found to perform better than the \xed" scheme. Both of these schemes rst calculate the base-r code as under the \xed" scheme. Then, if only one shortest path to the destination exists, the base-r value is used to route the message. However, if multiple shortest paths exist, the \partially balanced" scheme will choose the route code R 0 such that,ifr +((c d mod P ) M) <P h, then R 0 = R +((c d mod P ) M) else set R 0 = R. Whenmultiple paths exist, this approach spreads the trac across dierent links, based on the destination node number. However, if the number of shortest paths exceeds P, this approach limits its selection to the rst P such paths. The \random" scheme simply computes the number of alternate shortest paths from Eq. (2) and assigns

6 Iness, Banerjee, & Mukherjee: GEMNET 6 the route code R 00 = R +(M Z) where Z is a uniformly-distributed random integerin[1 Y) and Y is the number of shortest paths given by Eq. (2). Some representativenumbers for the link loading statistics are shown in Table 1. We defer discussing the results in this table until the following subsection. 3.2 Which Conguration of GEMNET is the Best? Multiple GEMNET congurations exist for a given number of nodes, so the question on which one has the lowest average hop distance or optimum link loading naturally arises. Since we do not have a closed-form solution for the average hop distance of a GEMNET, we cannot state elegantly which conguration of GEMNET will have the minimum average hop distance. However, based on the nature of the interconnection pattern, we can place some fairly tight bounds on the average hop distance. In a GEMNET, beginning at a node, all of the nodes reachable on a certain hop count belong to a specic column, and they are all contiguous within that column (since row indices wraparound). This allows a simple algorithm to determine the minimum and the maximum average hop distance from anynode. To determine the minimum average hop distance, we would like for the nodes covered on a given hop number to all be new nodes (i.e., nodes which have not been visited before). However, since GEMNET has K columns, at least K hops are needed before a node could possibly be revisited. The routing algorithm covers up to P i nodes (where i is the current hop number) that were not covered on previous passes within a column. This process is continued until all nodes have been covered. By keeping track of the number of nodes covered at each hop distance, up to the diameter D of the network, we can easily determine the minimum average hop distance as follows: minimum average hop distance = P Di=1 i r i M K where r i = min(p i M ; P bi=kc j=1 bp i;jk c 3 is the number of new nodes covered on hop i. To determine the maximum average hop distance we want tocover P i new nodes on hop number i when i<k,andcover up to P i ; P (i;k) new nodes on hop number i whenever i K. Let r 0 i = min(p i ;bp i;k c M;bP i;k c). Substituting r 0 i for r i in the above equation (viz. Eq. (3)), we solve for the maximum average hop distance. The largest dierence between the maximum and the minimum average hop distances in the worst case is one and this happens when P =2, K =1,and N(= M K) is 3 The P i;jk term represents the numberofnodescovered in the column on the j th previous pass. (3)

7 Iness, Banerjee, & Mukherjee: GEMNET 7 large. When K and P are small, there exists the largest asymmetry possible between the average hop distance for the minimum and the maximum cases. From the equations above, when K = 1 and P =2,wehave r i =min(2 i N ;b2 i;1 c;b2 i;2 c;b2 0 c) and ri 0 = min(2 i ;b2 i;1 c N ;b2 i;1 c). For 2 i<d, the ri 0 for the upper bound equals the r i;1 for the lower bound. For large N, these terms will dominate the r 0 0 term and the r D terms. Since ri 0 = r i;1, and since Eq. (3) multiplies the ri 0 and the r i terms by i, we see that the average hop distance of the two cases will dier by atmost1. Figure 2 demonstrates how the average hop distance in dierent GEMNET congurations compare with one another. This gure considers a 64-node GEMNET with nodal degree of 2. In general, the larger the number of columns (i.e., the \fatter" the GEMNET) is beyond an \optimal" conguration, the higher is the mean hop distance. In this example, the 2- column GEMNET exhibits the lowest hop distance, and it is superior, in this sense, to the corresponding 1-column GEMNET (a de Bruijn graph) because the latter has two nodes (top and bottom nodes) with links that transmit back to themselves (self loops), and these two nodes have larger individual mean hop distances (at most 1 greater) than the other nodes. The average hop distances for dierent GEMNET sizes and dierent values of K are shown in Fig. 3. The plots for K =2andK = 3 contain only the points where N is divisible by 2 and 3, respectively. In general, for a GEMNET with P =2andaneven number of nodes, K =2gives the minimum average hop distance. For a GEMNET with P =2,the minimum hop distance for a given N always has K 3 columns. For P = 2 and an odd number of nodes, in general, the lowest average hop distance occurs for GEMNETs with K = 1. Also, for the GEMNETs with P 3, in general the best average hop distance is achieved by a GEMNET with K = 1. Some of these results can also be seen in Table 1. In terms of link loading, we observe from Table 1 that, in general, the \random" routing scheme performs better than the \xed" and \partially balanced" schemes. Under the \xed" scheme, the 0-links out of a node would be utilized more often when multiple paths exists. For the \partially balanced" scheme, the higher-numbered routes would not be chosen due to the mod-by-p operation (e.g., if there were four shortest paths and P = 2, only the two lowestnumbered routes would be considered). \Random" routing eliminates this bias towards lowerest-numbered routes hence, it performs better. Henceforth, we limit our discussion to the \random" scheme only, but for additional results, see Table 1. Two opposing factors compete when we try to minimize the maximum link load. One is that, as a GEMNET is widened (increase K), a larger number of multiple shortest paths exist, allowing trac to be better balanced, thereby decreasing the load on the most-congested link.

8 Iness, Banerjee, & Mukherjee: GEMNET 8 On the other hand, as a GEMNET is widened, its average hop distance increases which will proportionally increase its average link load (= ((N ; 1) h)=p ). For the \random" routing scheme, we observe fromtable 1 that an increase in the average link loading tends to increase the load on the most-congested link. We observe that, for small N, the value of K that minimizes h also results in the minimum max-link load. (The optimum values of h, l i and (l i )max (for \random" routing), for a given (N,P) combination are italicized in Table 1.) However, as N gets larger this trend does not hold. Due to various factors on which link loading depends, we can only state that, for large N, the value of K that minimizes the maximum link loading is generally slightly larger than the K which minimizes h. 4 Scalability { Adapting the Size of a GEMNET This section investigates how to scale the size of a GEMNET. Specically, we examine dierent ways to grow a GEMNET. Approaches to decrease the size of a GEMNET are similar and are not included here to conserve space. See [7] for more information. In addition, our approaches to scaling a GEMNET apply to its implementation based on a single passive star, because a topology reconguration under this implementation can be easily performed by retuning some of the transmitters and/or receivers at the various nodes. Reconguring a multi-star [4] or switched implementation of GEMNET would be more involved. Since a one-column GEMNET can accommodate any number of nodes, there is always at least one interesting GEMNET (besides the one with P parallel rings corresponding to M = 1). From Section 3.2, we found that, for P = 2, the one-column and the two-column GEMNETs in general had the best average hop distance for odd and even number of nodes, respectively. This property isvery desirable for scalability since the one-column GEMNET can easily be grown by one node at a time (i.e., by adding an extra row), while the twocolumn GEMNET can be grown by two nodes at a time also one node can be added to a two-column GEMNET with 2j nodes (j =1 2 3 )by setting up the new structure as a one-column GEMNET with 2j + 1 nodes. Since, in general, the best average hop distance for P 3isachieved by a one-column GEMNET, there is a dual benet in that the most scalable structure also possesses the best average hop distance. The easiest way togrow a GEMNET is to add one node at the bottom of each of its columns. Thus, a GEMNET can grow by K nodes at a time (i.e., with modularity K). Adding K nodes in the bottom row turns out to be the best location to add nodes because the interconnection pattern is interrupted at the farthest point from the topmost row. Adding nodes closer to the top would have caused the interconnection pattern to be

9 Iness, Banerjee, & Mukherjee: GEMNET 9 interrupted earlier. Thus, to add one row of nodes, each node must determine what are the row numbers of the nodes that it needs to connect to in the new larger network and then retune its transmitters or receivers accordingly. An example of growing a (1,6,2) GEMNET by one node to a (1,7,2) GEMNET in this fashion is shown in Fig. 4. By examining the structure of a GEMNET, we observe that, when adding a row ofnodes, approximately the rst M nodesineach column need not retune. Approximately the next P M P nodes do one retuning, the next M nodes do two retunings, and so on. To be exact, when P adding K nodes to an N-node GEMNET in the fashion described above, we obtain number of retunings = PX i=2 N P (i ; 1) N (P ; 1) 2 with the equality holding when N is divisible byp. Since the total number of links (and hence transmitters/receivers) equals NP, for a P = 2 GEMNET, this means approximately 1 4 th of the total number of transmitters or receivers in the network need to be retuned, while for a P = 3 GEMNET, approximately 1 3 rd of the transmitters or receivers need to be retunned. Adding more than one row to a GEMNET can be performed in stages by adding one row at a time until all rows are added. Alternatively, the network can be scaled from the original setup to the nal setup in one massive retuning operation, by \optimally" \renumbering" the individual nodes in order to minimize the number of retunings. Deletion of a row of nodes from GEMNET can also be performed similarly. Space limitations prohibit us from discussing these details, but we refer the interested reader to [7] for more information. 5 Discussion Since multiple congurations of a GEMNET exist, one must consider its following properties before determining which conguration to choose: (1) scalability, (2) lowaverage hop distance, and (3) balanced link loading. Our recommended approach is to use a one-column GEMNET since it provides the maximum exibility innetwork sizes (i.e., good scalability property). For the P = 2 case, a one-column GEMNET will not give the best average hop distance for certain network sizes, but the average hop distance for a one-column GEMNET is very close to the best. The small benet in average hop distance for the other congurations is probably not worth the loss in exibility of network sizes. However, for the specic case of P = 2, a network with K = 2 has properties that may be benecial. It always has a lower average hop distance than a one-column GEMNET for an even number of nodes. For certain networks, the capacity of a link would be the determining factor in deciding which GEMNET to use. If, say, a one-column GEMNET has a maximum link load that

10 Iness, Banerjee, & Mukherjee: GEMNET 10 is higher than the maximum carrying capacity of a link, then a wider GEMNET (with \random" routing) could be used since this will reduce the maximum link ow. However, if link capacity is not a bottleneck, then we believe that scalability innetwork sizes is a much more important property and therefore we recommend the use of a one-column GEMNET. References [1] C. A. Brackett, \Dense Wavelength Division Multiplexing Networks: Principle and Applications," IEEE Journal on Selected Areas in Commun., vol. 8, no. 6, pp , Aug [2] S. Banerjee, Optimally-Structured High-Speed Metropolitan Area Networks with Distributed Control, Ph.D. Dissertation, U.C. Davis, Dec [3] I. Chlamtac, A. Ganz, and G. Karmi, \Lightnet: Lightpath based solutions for wide bandwidth WANs," Proc., IEEE INFOCOM '90, San Francisco, CA, pp , June [4] A. Ganz, B. Li and L. Zenou, \Recongurability of Multi-Star Based Lightwave LANs," Proc., IEEE GLOBECOM '92, pp , Aug [5] P. E. Green, \The Future of Fiber-Optic Computer Networks," IEEE Computer, vol. 24, no. 9, pp , Sept [6] M. G. Hluchyj and M. J. Karol, \Shue Net: An application of generalized perfect shues to multihop lightwave networks," J. Lightwave Technol., vol. 9, no. 10, pp , Oct [7] J. Iness, S. Banerjee, B. Mukherjee, \GEMNET: A Generalized, Shue-Exchange-Based, Regular, Scalable, and Modular Multihop Network Based on WDM Lightwave Technology," Technical Report no. CSE-94-8, Dept. of Comp. Science, Univ. of Cal., Davis, June [8] B. Mukherjee, \WDM-based local lightwave networks { Part I: Single-Hop systems," IEEE Network Magazine, vol. 6, no. 3, pp , May [9] B. Mukherjee, \WDM-Based Local Lightwave Networks - Part II: Multihop Systems," IEEE Network Magazine, vol. 6, no. 4, pp , July [10] R. Ramaswami and K. Sivarajan, \A Packet-Switched Multihop Lightwave Network Using Subcarrier and Wavelength Division Multiplexing," IEEE Transactions on Communications, vol. 42, no. 3, pp , March [11] R. Ramaswami, \Multiwavelength Lightwave Networks for Computer Communication," IEEE Communications Magazine, vol. 31, no. 2, pp , Feb [12] K. Sivarajan and R. Ramaswami, \Multihop lightwave networks based on de Bruijn graphs," Proc., IEEE Infocom '91, Bal Harbor, FL, pp , Apr [13] S.T. Tan and D.H.C. Du, \Embedded Unidirectional Incomplete Hypercubes for Optical Networks," IEEE Transactions on Communications, vol. 41, no. 9, pp , Sept

11 Iness, Banerjee, & Mukherjee: GEMNET 11 Appendix: Proofs That GEMNET Encompasses Other Structures To prove that a (K,M,P) GEMNET reduces to ShueNet when M = P K,we need to show that the following two conditions hold: (1a) we can arrange the N = KP K successive columns of P K users in K nodes each and (2a) numbering users in a column from 0 to P K ; 1 will cause node i to have arcs directed to nodes j, j+1,..., and j + P ; 1 in the next column, where j =(i mod P K;1 ) P [6]. Since the architecture of a GEMNET is set up with K columns of M nodes in each column, if M = P K,thenN = K M = KP K and therefore condition (1a) holds. By the denition of how a node within a column attaches to nodes in the next 4 column, we see that node r connects to nodes ^r, ^r + 1,..., and ^r + P ; 1 where ^r =(r P )modp K. Due to an identity, this last equation can be easily shown to be equivalent to^r =[r mod P K;1 ] P. This last equation is the same as (2a), but with dierent variables. Therefore, a (K,M,P) GEMNET reduces to ShueNet when M = P K. To prove that a (K,M,P) GEMNET reduces to a de Bruijn graph for M = P D and K =1, we need to show two conditions: (1b) the number of nodes in the GEMNET is N = P D for a positive integer D 2 and (2b) if the nodes are numbered in a base-d format, then node A =(a 1 a 2 ::: a D ) will connect to node B =(b 1 b 2 ::: b D )ib i = a i+1 a i b i ::: P ; 1 1 i<d; 1 [12]. Due to the constraints M = P D and K =1,thenumber of nodes is N = M K = P D, so condition (1b) holds. We can restate (2b) as: node (a 1 a 2 ::: a D ) will connect to nodes (b 1 b 2 ::: b D )=(a 2 ::: a D X) where X ::: P ; 1. In other words, node A connects to nodes ^b, ^b+1,..., and ^b+p ;1 where ^b =(amodp D;1 )P (where we note that the A mod P D;1 operation will leave (0 a 2 a 3 ::: a D ), which when multiplied by P will yield b =(a 2 ::: a D 0)). Due to the fact that ^b =(a mod P D;1 ) P is equivalentto^b =(a P mod P D ) (which is the formula for r with dierent variables), we see that condition (2b) is satised by GEMNET. Therefore, a (K,M,P) GEMNET reduces to a de Bruijn graph when M = P D, for a positive integer D 2, and K =1. Thus, GEMNET is a generalization of both ShueNet and de Bruijn graph. 4 Take into account \wraparound" of the last column to connect to the rst column.

12 Iness, Banerjee, & Mukherjee: GEMNET 12 0 λ0,λ2 λ10,λ11 Network Node λ10,λ12 λ0,λ1 1 2 λ4,λ6 λ12,λ13 Optical fiber λ14,λ16 λ2,λ3 3 4 λ8,λ1 λ14,115 PASSIVE STAR λ18,λ11 λ4,λ λ3,λ5 λ16,λ17 λ7,λ9 λ18,λ19 Wavelengths transmitted Wavelengths to be filtered from all wavelengths λ13.λ15 λ6,λ7 λ17,λ19 λ8,λ9 (a) Physical topology & transciever tuning pattern link 1-link 0,0 Node 0 λ0 1,0 λ10 0,0 λ1 Node 1 λ11 Node 0 λ2 0,1 1,1 Node 2 λ3 Node 3 0,2 Node 4 λ4 λ5 1,2 Node 5 0,3 Node 6 λ6 λ7 1,3 Node 7 0,4 λ8 1,4 λ9 Node 8 Node 9 λ12 0,1 λ13 Node 2 λ14 0,2 λ15 Node 4 λ16 0,3 λ17 Node 6 λ18 0,4 λ19 Node 8 i-link from Node at (c,r) connects to Nodes at ((c+1) mod 2, (2r+i) mod 5), i=0,1. (b) Logical structure (virtual topology) corresponding to the transceiver assignments in (a). Figure 1: A 10-node (2,5,2) GEMNET.

13 Iness, Banerjee, & Mukherjee: GEMNET Bounds MEAN HOP DISTANCE (de Bruijn graph) (ShuffleNet) K Figure 2: Bounds and average hop distance for a P=2, 64-node GEMNET with dierent values of K.

14 Iness, Banerjee, & Mukherjee: GEMNET h avg. hop with P=2, K=1 avg. hop with P=2, K=2 avg. with hop P=2, K= N Figure 3: Average hop distance for dierent N and K.

15 Iness, Banerjee, & Mukherjee: GEMNET 15 Scaling (Growing) (1,6,2) GEMNET (1,7,2) GEMNET 0,0 0,0 Node 0 Node 0 0,1 0,1 Node 1 Node 1 0,2 0,2 Node 2 Node 2 0,0 0,0 Node 0 Node 0 0,1 0,1 Node 1 Node 1 0,2 0,2 Node 2 Node 2 retuned (3) pretuned (2) nontuned (9) (original wavelength assignment) 0,3 0,3 0,3 0,3 Node 3 Node 3 Node 3 Node 3 0,4 0,4 0,4 0,4 Node 4 Node 4 Node 4 Node 4 0,5 0,5 0,5 0,5 Node 5 Node 5 Node 5 Node 5 0,6 0,6 Node 6 Node 6 (a) Original (1,6,2) GEMNET (b) New (1,7,2) GEMNET Figure 4: Growing a (1,6,2) GEMNET by one node.

16 Routing Part. Bal. Rounting Random Routing Fixed N K Network Avg. Hop D Std. dev. Avg. link Std. dev. (li)max: Std. dev. (li)max: Std. dev. (li)max: P GN SN DB dist. (h) of hi load (l) of li (li)min of li (li)min of li (li)min X X : : :7 2 X X : : :4 4 X : : : X : : :15 2 X : : :16 3 X X : : :26 4 X : : : X X : : :63 2 X : : :44 4 X X : : :116 8 X : : : X X : : :255 2 X : : :183 4 X : : :562 8 X : : : X X : : : X : : :749 4 X : : : X : : : X X : : : X : : : X X : : : X : : : X X : : :85 2 X : : :92 4 X : : :218 8 X : : :314 1: Comparison of GEMNET (GN), ShueNet (SN) and de Bruijn (DB) graph. (Link loads are computed under the assumption of Table unit of ow between every source-destination pair and dierent routing schemes. Also in this table, h is averaged over all the individual one nodes' average hop distance (hi).)