The Multi-dimensional Shuffle-exchange Network: A Novel Topology for Regular Network Architectures *

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1 The Multi-dimensional Shuffle-exchange Network: A Novel Topology for Regular Network Architectures * Philip P. To Tony T. Lee Department of Information Engineering The Chinese University of Hong Kong Shatin, N, T,, Hong Kong ttlee@ ie.cuhk.edu.hk Tel: Fax: Abstract In this pape~ a novel class of network topologies known as the Multi-dimensional Shu@e-exchange Network (A4DSXN) is proposed. We show that the well-known de Bruijn graph and hypercube in fact both belong to the same class of graphs represented by MDSXN. The MDSXN is therefore the unification and generalization of the de Brutjn graph and hypercube. We show that members of MDSXN inherit the topological properties of both the de Bruzjn graph and hypercube to a varying degree. This allows us to tradeoff cost andperformance effectively and construct networks which are most suitable for a particular purpose. Examples of applications of MDSXN include structure for switching and multicasting networks, optical network topology, and virtual topology for local and metropolitan area networks. I. Introduction Interconnection network plays a very important role as architectures of multiprocessor computers, fast packet switches and optical networks. In general, the properties of a network are largely affected by the topological properties of the underlying graph, with vertices representing the network nodes and edges representing the links. It is therefore important to select a graph which possesses properties most suitable for the particular application. Two well-known graphs which have been widely studied are the de Bruijn graph and the hypercube. The de Bruijn graph is a regular graph and is characterized by two parameters d and n. A (d, n) de Brttijn graph network consists of N = dn nodes, each with in-degree and out-degree equal to d. In the binary de Bruijn graph (d = 2), each node is labeled by an n-bit address. Nodal connectivity is determined by the shuffleexchange operation (a left-shifting operation plus a new input bit). there is a directed edge. from node (xoxl... xn _ 1) to nodes (x1x2.xn_lo) and (X1X2... Xn 11). The de Bruijn graph has the unique-path self-routing property and the path from a source *This work wassupported by the ResearchGrant Council of Hong Kong under Eza markedgrant No. CUHK 4 177/97E. to a destination can be completely determined from the source and the destination addresses. The diameter of the graph is equal to n. It has been shown that the connectivity of a (d, n) de Bruijn graph is d 1 [1] and it can tolerate up to d 2 node failures. Tlhe de Bruijn graph is known to have low mean internodal distance approaching the Moore limit [2]. Multistage realization of the de Bruijn graph yields the shuffle-exchange network. An example (2,3) de Bruijn graph network is shown in Fig. l(a). (a) 000,oo@/@~> ]*+J lol@/ 17:@/ Figure 1: (a) A (2,3) de Bruijn graph network; (b) A 3-dimensional hypercube network. An n-dimensional hypercube is a regular graph with N = 2 nodes. Each node is labeled by an n-bit binary address. Two nodes are connected by a hi-directional link if their addresses differ by one and only one bit. The in-degree and out-degree of a node are both equal ton. The distance between two nodes is equal to the Hamming distance of their addresses, which implies that the diameter of the graph is equal to n. Self-routing is possible. It has been shown in [3] that between any pair of nodes in the hypercube, there are n node-disjoint parallel paths between them. In other words, hypercube can tolerate up ton 1 node failures without becoming disconnected. Because of this strong fault tolerant ability, hypercube has found applications in multiprocessor computers where reliability is critical. An example 3-dimensional hypercube is shown in Fig. l(b). For a given number of nodes N, the de Bruijn graph has a fixed node degree of 2 while the hypercube has a node degree of (b)

2 log N. In general, the implementation cost of a network is directly proportional to the node degree. The higher node degree of the hypercube leads to a better performance and fault tolerance over the de Bruijn graph, in which there is only a unique path between any pair of nodes and cannot tolerate any failure. Therefore, we can observe a tradeoff between cost and performance, Between de Bruijn graph and hypercube, it is desirable to find graphs which possess the properties of both the de Bruijn graph and the hypercube and with intermediate cost and performance, In this way, we can tradeoff cost and performance more effectively and construct networks which are most suitable for a particular application. In [4], for example, the authors define a class of networks called the Hyper-deBruijn networks. The address of each node in a HyperdeBruijn network is a combination of a de Bruijn graph address and a hypercube address, known respectively as the debruijnpart-label and a hypercube-part-label. In this paper, we show that both the de Bruijn graph and the hypercube in fact belong to the same class of networks which we denote it as Multi-dimensional Shuffle-exchange Network (MDSXN). Member networks in the family possess properties of both the de Bruijn graph and the hypercube to a varying degree. We will study the various topological properties of MDSXN. The MDSXN can be used in a wide range of applications such as switching and multicasting networks, and topology for multihop optical networks [5]. Moreover, due to its flexible structure, the MDSXN is also suitable as virtual topology for routing in local and metropolitan area networks [6, 7]. II. Structure The multi-dimensional Shuffle-exchange Network (MDSXN) has a toial of N = 2 nodes, where n is the number of addressbits. The adclress bits are partitioned into k components numbered O, 1,... k --1. Each address component corresponds to a dimension. Letm = [mornl.. m&1 ] be a row vector and let mi be the number of bits in partition i. The vector m is called the partition vector and we have ~~~~ mi = n. An address X is of the form x = (X(j,xl,...x1)l) = (Xo... %no 1, %...xmo+ml l.... ) ~~ no ml The net work can be characterized by the three-tuple (n, k, m). Considm an (n, k, m) MDSXN. Connectivity between nodes is determined by the shuffle-exchange operation on each address component. There is a directed edge connecting a node A to a node B if one address component of B can be obtained from the corresponding component in A by one shuffle-exchange operation. For example, in a 2-dimensional MDSXN, a node having address (xox 1. Xmo 1> xmoxmo+l... Xmo+m,-l) is connected to the nodes with the following addresses: (X1X2 ~.. X~o_l O, x~ox~o+l.x~o+tn l ) (,X1X2.,Jfno-ll, %noxmf)+l Xmo+lnl-l); (Xoxl.,.Xrno-l, XrnO+lx~O+2 Xwzo+ml-1 O); and (Xoxl...X~o_l,XrnO+lx~O+2. Xmo+ml 11). The former two are due to shuffle-exchanging in the first dimension while the latter two are due to shuffle-exchanging in the second dimension. Since either a O or 1 can be appended during each shuffle-exchange operation, the number of outgoing links in each node is equal to the number of incoming links and is equal to 2k. The MDSXN is multihop in the sense that a packet may need to go through a number of intermediate nodes before reaching its destination. It is easy to see that the diameter of the network, which is the maximum distance between any pair of nodes, is equal to n. Figure _ shuffle-exchange in first component ~ shuffle-exchange in second component 2: A (3, 2, [1 2]) MDSXN. Figure 2 shows a (3,2,[1 2]) MDSXN. In this network, each node has 4 outgoing links and 4 incoming links. We can see that some nodes have self-loops. A node has a self-loop if one of the address components contains either all O s or all 1 s. The number of self-loops of a node is equal to the number of address components which contain all O s or all 1 s. When MDSXN is used as, say a closed switching networks, these self-loops may become redundant and may be eliminated. Both the de Bruijn graph and the hypercube belong to the same family of networks represented by MDSXN. Consider an (n, k, m) MDSXN and consider two special cases: k = 1 and k = n. When k = 1, the entire n-bit address is treated as a single component and the partition vector can only be equal to [n]. Since connectivity in each dimension is determined by the shuffle-exchange operation, the (n, 1, [n]) MDSXN degenerates to a (2, n) de Bruijn graph network. When k = n, each address component consists of a single bit and the partition vector can only be equal to [ ]. In this case the network becomes an (n, n, [ ]) MDSXN and it degenerates to a hypercube. Each dimension in the hypercube network corresponds to a component of 1 bit in size in an MDSXN. The node degree of a hypercube is therefore equal to 2n instead of n, but n of which result in self-loops. It is not obvious that nodes in a hypercube in fact have self-loops. This is because these selfloops are redundant and are usually omitted when hypercube is

3 Multi-dimensional Shuffle-exchange Network n bits, k components Xo Xmo l, xmo...xmo+m, -l,...,xmo+ml+,,, +mk_2...xn_l -~ ~ mo ml ink-l la= z@~ xn ~ Xo, xl,....x] ] n bits, 1 component n bits, n components (de Bruijn graph) (hypercube) Figure 3: Both the de Bruijn graph network and hypercube network belong to the same family of networks represented by MD- SXN. used in switching networks or multiprocessor computers. The two cases are illustrated in Fig. 3. Given the number of address bits n, the de Bruijn graph and the hypercube are the two extreme cases in the family of networks represented by MDSXN. By varying k and the partition vector m, we obtain a range of intermediate networks with different characteristics. Intuitively we can expect that these intermediate networks should inherit the properties of both networks to a varying degree. It will become apparent in later sections that this is indeed the case. III. Routing in MDSXN Self-:routing in MDSXN is performed by means of a routing tag. The routing tag contains the bits that are inserted in each shuftle-exchange operation for each component and can be obtained from the source and the destination addresses, To compute the routing tag, each component in the source address is compared with the corresponding component in the destination address to determine the routing bits required. Consider an (n, k, m) MD- SXN. Let the source address be S = (S0, S1,.... S~_l) and the destination address be D = (Do, D1,.... Dk_l), where Si s and Di s are address bit sequences corresponding to dimension i. By our nota,tion, Sj and Dj can be written as Sj = mo+ml+...+ml l Smo+m~+...+m1,l, and Dj = dmo+ml+. +mj-l... dmo+ml+...+ml l. routing tag = dmo+m~+... +mi i dl?lo+~l++)?tj-l hop distance hj = i else routing tag = NULL hop distance Itj = O end if The function compare (aoal... an l, bobl... bn_l, i ) is TRUJ3ifaj+i =bjforall~where O~~ <n i l. The routing tag is empty (which we denote by NULL) if the corresponding components are identical in the source and the destination addresses. The algorithm basically compares the source and the destination address components and successively left-shifting the source address component and right-shifting the destination address component until they match. When they do, the bits that have been shifted out in the destination address component constitute the routing tag for dimension j. In general, the routing tag is of the form (ro., rho 1, rho...rho+hl_l..... ~~ ho hl rho+hl+...+hz-z \ ~ mo+hl+... +hk_l-l ) hk- 1 where hj (O s j < k 1) is the number of routing bits in component j, which is also equal to the number of shuffle-exchange operations required to transform Sj to Dj. It is easy to see that hj ~ mj for all j where O ~ j s k 1, As an example, in a (5,2,[2 3]) MDSXN, to go from node (01,010) to node (10,001), the routing tag is (0,01). The hop distance in each dimension is equal to the length of the routing tag in the corresponding component. In this example, the hop distances are 1 and 2 in the first and the second dimension respectively. The state of a packet in MDSXN can be represented by the two-tuple (X; R), where X is the address of the node that the packet currently resides and R is the routing tag. From one state a packet may jump to one of the several possible states. For the sake of illustration, consider an (n, 2, m) MDSXN. With the self-routing algorithm, the state transition diagram of a packet in node (XO~.. Xmo 1,xmo... ~mo+ml 1) with routing (ro... rho l, rho -. r~o+~l _ 1) is given by tag The fol [owing algorithm computes the routing tag for component j by comparing Sj and Dj. This algoritlmn is similar to the shortest-path routing algorithm illustrated in [8] for de Bruijn graph networks. procedure compute-routing-tag(j) i=() while compare (Sj, Dj, i) Q FALSE i=j+l end while if i # O then where the notations < ro, * > and < *, rho > indicate that routiw is performed in the first and the second dimension respectively. The packet may be routed in any dimension as long as the corresponding component in the routing tag is non-empty. It is easy to see that the number of possible regular transitions is equal to the number of non-empty components in the routing tag. This implies

4 that within a node, a packet may have several desired output ports to go to and it needs only be routed to any one of them. After each transition, the routing bit used is removed from the routing tag. The packet will reach its desired destination when the routing tag is exhausted. Note that in each routing step, a packet may be routed in any dimension as long as the corresponding component in the routing tag is non-empty. This implies that there may exist multiple alternative shortest paths from one node to another. As an example, Fig. 4 shows the three alternative shortest paths from node (01,010~ to node (10,001)ina(5,2,[23]) MDSXN. In this example, the routing tag is (0,01) and the length of the shortest path is 3. In the first routing step, the packet may be routed in the first dimension or in the second dimension. Since there is only 1 routing bit in the first dimension, after the packet is routed in the first dimension, that routing bit is dropped and the packet can only be routed in the second dimension in subsequent steps. If the packet is routed in the second dimension we have the choice of routing in the first step, then the packet in the first or the second dimension in the next step. We can see from the figure that the number of alternative shortest paths is equal to 3. In general, the number of alternative shortest paths from one node to another in an (n, k., m) MDSXN is equal to the multinominal coefficient Note that these alternative shortest paths are not node-disjoint. For de 13ruijn graph, we have k = 1 and the number of alternative paths is equal to ~ = 1. For hypercube, we have k = n and O<hj<lfor alljwhere O<j S k l. The numberof alternative paths is therefore equal to (~~~~ h j )!. <0,.> /EEP--E& E 01,010 \,,,,, (0,01) <,,1> n (NULi,NULL) 10,100 <o, > <,0> (NULL,I) KQy 01,100 (0,1) <0, > <,1> 01,001 (o,null) Figure 4: Three alternative shortest paths from node (01,010) to node (10,001). IV. Topological Properties of MDSXN In th~s section, we investigate various topological properties of MDSXN, including the mean internodal distance, the mean effective nocle degree, existence of node-disjoint paths and existence of cycles and Hamiltonian circuits, (1) A. Mean Internodal Distance In MDSXN, a packet may need to pass through multiple intermediate nodes to go from the source to the destination. The mean internodal distance is the average length of the shortest path between every pair of nodes in the network. The mean amount of time a packet stays in the network is directly proportional to the mean internodal distance of the network. If the mean internodal distance is large, on average a packet will need to stay longer in the network, taking up valuable resources such as bandwidth and buffers. By Little s Law, this will lead to a lower throughput asnew packets may not be able to enter the network, This effect is more pronounced in networks with large propagation delay. Therefore, the performance of an MDSXN, in terms of throughput and delay, improves with the decrease of mean internodal distance, In this section, we will compute the mean internodal distance for an (n, k, m) MDSXN. Consider two nodes A and B in an (n, k, m) MDSXN. The hop distance to go from A to B is equal to the number of shuffleexchange operations required to transform the address of A to that of B, More rigorously, if it takes i hops to go from node A to node B, we have ~. 1h j = i, where h j is the number of routing bits in dimension j. In general, the hop distance from A to B may not be equal to the hop distance from B to A. In other words, the hop distances between two nodes may not be symmetric. For example, in a (3,2,[ 12]) MDSXN, it takes 3 hops to go from node (1,01) to node (0,00) (1 hop in the first dimension and 2 in the second) but it only takes 2 hops to go from node (0,00) to node (1,01). The hop distances are therefore not symmetric. Consider all possible source-destination pairs in an (n, k, m) MDSXN. Let O(i) be the number of source-destination pairs in which the shortest distance from the source to destination is i hops. The total number of source-destination pairs is N(N 1), Since the maximum hop distance from one node to another is n, the mean internodal distance is equal to [8] (2) The following method to compute the mean internodal distance of an MDSXN is a generalization of the method to compute the mean internodal distance of a de Bruijn graph network outlined in [8]. As mentioned, if we consider the subnetwork formed by fixing all but one address components of j bits, the subnetwork is a binary de Bruijn graph network with 2~ nodes. Let di (j ) be the number of source-destination pair in a binary de Bruijn graph network with 2j nodes (number of address bits= j ) in which the shortest distance from the source to the destination is i hops, where i ~ () and j ~ 1. A node is considered to be zero hops away from itself and do(j) = 2j. The values of di (j ) can be obtained by a method outlined in [8] and O(i ) can be computed by

5 -- k 1 x ~dhj(mj). (3) To explain (3), suppose it takes i hops to go from node A to node B, There are many ways to distribute the hop distance into different components, provided that they all sum up to i. Basically (3) computes the number of source-destination pairs with hop distance i for all possible combinations of h ~ s. Table 1 shows the mean hop distance for the MDSXN with different values of n, k and m. The partition vector m is shown along with the mean internodal distance whenever there are multiple ways to partitior~ the address bits, Table 1: Mean internodal distance of an (n, k, m) MDSXN n k , [1 2] 2.50-[1 3] 3.26-[14] 4.10-[1 5] 1.86-[2 1] 2.40-[22] 3.06-[23] 3.84-[24] 3.74-[33] [1 1 2] 2.94-[1 1 3] 3.71-[1 14] 2.27-[1 2 1] 2.84-[1 22] 3.52-[1 23] 2.27-[2 1 1] 3.43-[22 2] [1 11 2] 3.40-[1 11 3] 3.30-[1 1 22] [ ] A number of observations can be drawn from Table 1. each routing step, only nodes which are not yet visited will be reached. Obviously this is impossible. In an MDSXN, the nodes with one or more address components containing all O s or all 1 s have self-loops. These self-loops are redundant as far as mean internodal distance is concerned. A node with a lot of self-loops will reach fewer unvisited nodes in each routing step, leading to a larger mean internodal distance, For instance, Fig. 5 shows the routing trees of the node (100100) in an MDSXN with n = 6 and k = 3 for two different partition vectors rna = [1 14] and rnb = [22 2], In both trees, those nodes which have already been visited are omitted, We can see from the figure that for ma, the number of nodes visited after one hop is 4 while for mb, the number of nodes visited is 5. This is due to the fact that if rna is used, the first and the second components are both 1 bit long and two self-loops will be formed. These self-loops reduce the number of new nodes visited. After two hops, the number of nodes visited is 9 for rna and 11 for rnb. Therefore with rna it takes more hops for the node to visit all other nodes in the network, leading to a larger mean internodal distance. In addition, it generally takes more shuffle-exchange operations to loop a bit sequence back to itself if the sequence is long. Therefore to minimize the mean internodal distance, which is equivalent to maximizing the number of new nodes visited in each routing step, the size of each component should be maximized, which implies that all the partitions should be of equal size ,., For particular values of n and k, there are different ways to partition the address bits. If a certain way of partitioning is simply a permutation of another, their mean internodal distances will be equal, For example, when n = 4 and k = 3, partitioning the bits into [1 12],[121] and[21 1] all give the same mean internodal distance. As far as mean internodal distance is concerned, these networks can be considered as equivalent. For fixed value of n, the mean internodal distance is largest for k = 1 (a de Bruijn graph) and smallest for k = n (a hylpercube). This is because the node degree of a de Bruijn graph is 2 while the node degree of a hypercube is 2n. The larger the node degree is, the smaller the mean internodal distance will be, For networks with k ranging between 1 and n, their node degree ranges from 2 to 2n and their mean intemodal distance is upper bounded by that of the de Bruijn graph and lower bounded by that of the hypercube. For a fixed value of k, the mean internodal distance is minimized if all partitions are of equal size. For instance, if n = 6 an,~ k = 3, there are three different ways to partition the address bits - [1 2 3], [2 2 2] and [1 1 4]. The mean internodal di~;tance is smallest for the caseof[22 2]. The reason for this is intuitive. The mean internodal distance is minimized if in 0,0,0100 1,0,1001.., 1,1,1000 1,1,1001 1,0,0000 1,0, ,01,00 T-K, 00,10,00 00,11,00 10,10,00 10,01,11 10,01,10. A.. - / \ 01,,1!00 10,11, ,00,00 10,10,01 : ,01,00 01,10,00 01,01, Figure 5: Routing tree for node (100100) with rna = [114] and rnb = [222] respectively. B. Mean Effective Node Degree partition vector In an (n, k, m) MDSXN, the number of incoming links of a node is equal to the number of outgoing links and is equal to 2k. The number of self-loops of a node is equal to the number of address components which contain all O s or all 1 s. If shortest-

6 path routing is used, these self-loops are redundant and can be eliminated. We are therefore interested indetermining the mean effective node degree of an MDSXN. The mean effective node degree is defined as the average number of outgoing links of a node that do not loop back to the node itself. Since the effective incoming degree of a node in MDSXN is equal to its effective outgoing degree, the mean effective node degree also indicates the mean number of incoming links which are not formed from self-loops. no nodes in common except the source node and the destination node. The existence of node-disjoint paths is important because it provides a way of choosing an alternative path in case of node or link failure. The set of node-disjoint paths may also be used simultaneously as parallel paths to increase throughput. Theorem 1 If the addresses of two nodes A and Bin an (n, k, m) MDSXN differ in i dimensions, there will be i node-disjoint shortest paths between node A and node B. Table 2: Mean effective k [1 2] 2,50-[2 1] [1 3] 3.00-[22] 3.50-[1 1 2] 3.50-[1 2 1] 3.50-[21 1] 400 node degree of an (n, k, m) MDSXN [14] 2.94-[1 5] 3.25-[23] 3.38-[24] 3.74-[33] 3,75-[1 1 3] 3.88-[1 14] 4.00-[1 22] 4.25-[1 23] 4.50-[22 2] 4.50-[1 11 2] 4.75-[1 11 3] 5.00-[1 1 22] [ ] 6.00 Table 2 shows the mean effective node degree of an (n, k, m) MDSXN for different values of n, k and m. The following can be observed. c For a fixed value of n, the mean effective node degree of the de Bruijn graph (k = 1) is equal to 2 and it is equal to n for the hypercube (k = n), For MDSXN with intermediate values of k, the mean effective node degree is upper bounded by that of the hypercube and lower bounded by that of the de Bruijn graph. For particular values of n and k, there are different ways to partition the address bits. If a certain way of partitioning is simply a permutation of another, their mean effective node degrees are equal. For particular values of n and k, the mean effective node degree is relatively larger if the address bits are divided into equal parts. With reference to the corresponding values of mean internodal distance in Table 3, it can be seen that for a given n, the mean internodal distance decreases with the increase of the mean effective node degree. In general, it is more costly to build networks with large mean effective node degree. Therefore, be varied to tradeoff given n, the values of k and m can cost and performance. Node-disjoint Paths and Fault Tolerance In Section III we show that there exists a number of alternative paths between two nodes. These paths, however, are not necessarily node-disjoint. Two paths are node-disjoint if the paths contain Proofi Without loss of generality, assume that the addresses of the two nodes differ in the first i dimensions. We are going to show by construction that there are i node-disjoint shortest paths from A to B. Note that the choice of the set of node-disjoint shortest paths may not be unique. Let there be h ~ bits in component j of the routing tag, where O ~ j < k 1. By our assumption, since the two addresses only differ in the first i dimensions, we have hj=ofori~j~k l. In each routing step, a packet may be routed in any dimension as long as the routing tag is non-empty in that dimension, A path is uniquely determined by the sequence indicating which dimension it has gone through in each routing step. Two path will merge together if at some step their path sequences are permutations of one another. For example, to go from node (10,10) to (11,11), the routing tag is (11,11). Among all the possible paths, two of them are represented by the sequences 0101 or 1010, where a O indicates a hop in the first dimension and a 1 indicates a hop in the second dimension, The two paths will merge after two hops (i.e., at node (01,01)) asthe subsequences 01 and 10 are permutations of one another. Therefore to construct i node-disjoint paths from A to B, we must ensure that in each routing step, their corresponding path sequences will not become permutations of one another. The node-disjoint paths can be constructed as follows. From A we can branch out to i different paths by routing in each of the i dimensions. In subsequent steps, path i will keep routing in dimension i until all the routing bits have been used. Afterwards, routing is done in the next dimension (modulo i). It is not difficult to see that the paths formed by this routing method are nodedisjoint. No path sequences can be permutations of one another at any step because they all skirt from a different dimension. All the paths will finally merge at the destination node when all the routing bits are used, By a similar method, we can show that there are at least k node-disjoint paths (including those which are not shortest paths) between any pair of nodes. The existence of node-disjoint paths also has an implication on fault tolerance of MDSXN. In computer networks and distributed systems, network reliability is very important. It is desirable for the network topology to be fault tolerant. A fault tolerant topology IS able to maintain reasonable performance in the presence of node or link failures. In an (n, k, m) MDSXN, there are at least k node-disjoint paths between two nodes. The connectivity

7 of MDSXN therefore is equal to k and a network based on an (n, k, m) MDSXN can tolerate up to k 1 node or link failures. D. Hamiltonian Circuits and Cycles in MDSXN A Hamiltonian circuit in a graph is a closed path which visits each and every node in the graph exactly once [9]. A graph is Hamiltonian ifitcontains ahamiltonian circuit. In this section, wewillshow that MDSXN is Hamiltonian. Apart from being an interesting property, the existence of Hamiltonian circuits in MD- SXN is meaningful when MDSXN is used in certain applications. Forinstance, if MDSXN isusedas a multiprocessor architecture, certain algorithms may require data structure inform of a cycle oralinear array forefticient execution [3]. With the existence of Hamiltonian circuits, acycieor alinear array ofsize2n can be easily mapped onto MDSXN. Previous studies have shown that Hamiltonian circuits exist in de Bruijn graphs [10]. In other words, an MDSXN with only one dimension is Hamiltonian. For example, a Hamiltonian circuit in the (2, 1, [2]) MDSXN (or equivalently a (2,2) de Bruijn graph) is given by , For the sake of illustration, we will illustrate how a Hamiltonian circuit can be constructed in a 2-dimensional (n, 2, [rno ml]) MDSXN. Denote the dimension having a larger number of address bits as the high-order dimension. Without loss of generality, assume that mo? m 1 and the first dimension is the high-order dimension. A Hamiltonian circuit in the (n, 2, [m. m 1]) MDSXN can be constructed asfollows: An arbitrary node is chosen asthe originating node. By fixing the high-order dimension and transiting only in the low-order dimension, the subnetwork is equivalent to a de Bruijn graph with 2m1 nodes. We travel in this subnetwork by following the Hamiltonian circuit in the corresponding de Bruijn graph until it is one hop from completion (otherwise we will run into anode which has already been visited). After that we advance in the high-order dimension by one hop to enter another subnetwork. The high-order dimension is again fixed and we again travel in the low-order dimension, The procedure is repeated until all the nodes have been visited. As an example to illustrate the procedure, Fig. 6 shows how a Hamiltonian circuit can be constructed in the (4,2,[2 2]) MDSXN by starting with node (00,00). The first dimension is chosen as the high-order dimension. In the beginning, the first dimension is fixed and we successively transit in the second dimension along the Hamiltonian circuit of a (2,2) de Bruijn graph (since m 1 = 2). After three hops, we advance in the first dimension by one hop and enter another subnetwork. The procedure is repeated until all the nodes have been visited. Due to the structure of the procedure and alignment requirements, the number of address bits corresponding to the high-order dimension must be greater than or equal to the number of address bits in the low-order dimension; otherwise a Hamiltonian circuit cannot be constructed. This requirement, however, is not a problem as we can always find a dimension with more address bits as the high-order dimension. To construct a Hamiltonian circuit in an (n, k, m) MDSXN for k z 3, the above procedure can be recursively applied. We 00,00 00,01 00,11 00,10 01,10 01,00 01,01 1+ b I m10,00 b01,11 t 10,10 10,11 10,01 11,01 11,00 11,10 11,11 d+ ----~ transition in dimension O _ transition in dimension 1 Figure 6: A Hamiltonian circuit in a (4,2,[2 2]) MDSXN. can treat k 1 dimensions together as the high-order dimension and the remaining dimension as the low-order dimension. The only requirement is that the total number of address bits in the collaborated high-order dimension must be greater than or equal to the number of address bits in the low-order dimension. In the case of the (n, n, [ ]) MDSXN (a hypercube), the above procedure generates a Hamiltonian circuit with addresses corresponding to the set of n-bit Gray codes, which agrees with previous results on hypercube [3]. A Hamiltonian circuit in an (n, k, m) MDSXN corresponds to a cycle of length 2. In general, it is more difficult to map cycles of arbitrary length onto MDSXN, For a one-dimensional (n, 1, [mol) MDSXN, cycles of arbitrary length C, where 2 s C ~ 2m0, can be mapped. For MDSXN with a higher number of dimensions, it may not be possible to map cycles of arbitrary length onto the network. We can, however, determine whether a cycles of a particular length can be mapped onto the network from the parameters n, k and m = [mo, m 1,.... m&~]. Unfortunately, no closed-form solution exists. We will demonstrate the method by means of an example. Consider mapping a cycle of length C onto a 2-dimensional (n, 2, [mo ml]) MDSXN. Note that finding a cycle of length C is equivalent to finding a path which starts at an originating node and loops back to that node in C hops. From the originating node, the path may be constructed by routing in a number of different dimensions, If we focus on an individual dimension, since we require the path to loop back to the originating node, one or more cycles must be formed in that dimension; otherwise a closed path will not be formed. This implies that for each dimension, there must be a certain number of steps within the C hops which corresponds to forming one or more cycles in that dimension. Without loss of generality, assume that mo > m 1. Apart from the trivial case where C = 1, there are two cases: Case 1:2 ~ C ~ 2m0 + 2ml. A cycle can be mapped if C can be expressed as the sum of two integers co and c1: In that case, the cycle consists of co steps in the first dimension and c1 steps in the second dimension, or vice versa. For instance, a cycle of length 7 in the (4,2, [2 2]) MDSXN can be formed by routing 3 hops in the first dimension and 4 hops in the second, as

8 indicated in Fig. 7, By focusing on each dimension individually, it can be seen that the 3 hops in the first dimension result in one cycle in the first dimension ( ~ 00) and the 4 hops in the second dimension result in one cycle in the second dimension ( ). In each individual dimension i, we can construct a cycle with length ranging from 2 to 2ni. Therefore C can at most be equal to 2m0 + 2nl. transition in dimension O ---- transition in dimension 1 Figure 7: A cycle of length 7 in the (4,2,[2 2]) MDSXN. Case 2: 2rn0 + 2ml < C ~ 2. A cycle can be constructed by a method similar to the one used to construct the Hamiltonian circuit. Denote the dimension with a larger number of address bits as the high-order dimension (which is the first dimension in this example). As stated previously, each hop in the high-order dimension can be considered as entering another subnetwork. Within each subnetwork, a cycle can be formed in the low-order dimension as long as it does not run into a node which visited. If C can be expressed as h 1 has already been c=~ci+(h+l)+d (5) t=0 where 2<ci<2ml, 2<h<2m0 land O<d~2m0 (h+l), then the cycle can be mapped onto the 2-dimensional MDSXN. To interpret (5), h is the number of subnetworks entered and Ci is the length of cycle traveled in subnetwork i, where O ~ i ~ h 1. As an example, we can construct a cycle of length 10 in a (4, 2, [2 2]) MDSXN by choosing h = 2, co = 3, c1 = 4 and d = O. In other words, we traverse a cycle of length 3 and a cycle of length 4 in the two subnetworks respectively, as shown in Fig. 8. In (5), the special case where h = 2m0 is excluded. When h = 2~0, due to alignment requirements, we require that Co= cl=... = ch 1 = t and h needs to be a multiple of t. In that case, the cycle length C can be simply expressed as C = ht. One example of this special case is the construction of Hamiltonian circuit by choosing t = 2m1 and h = 2m0. The requirement of h being a multiple of k implies that mo > m 1. z00,00-b - 10, E3--zkz 10,11 01,11 1 I - 01,01 For an (n, k, m) MDSXN with k > 3, we can treat k 1 dimensions together as a unit and recursively apply (4) and (5) to determine whether a cycle of a certain length can be mapped. However, since both equations are not closed-form, it may require a lot of computations. As a final example, consider a (3,3,[ 11 1]) MDSXN. Since there is only 1 bit in each dimension, h = 2 and ct = 2 for all i. Therefore, no cycle of odd length can be mapped. It can be shown that all cycles of even length from 2 to 8 can be mapped. This agrees with previous results on hypercube [3]. V. Conclusions In this paper, a novel class of network topologies called the Multi-dimensional Shuffle-exchange Networks is proposed. We showed that the well-known de Bruijn graph and hypercube both belong to this class of graphs. The structure of MDSXN is very general and can be applied in different areas such as architecture for multiprocessor computers, packet switches and optical networks. We discussed the shortest-path self-routing algorithm and studied various topological properties of MDSXN including the mean internodal distance, the mean effective node degree, the existence of node-disjoint paths and the existence of cycles and Hamiltonian circuits. We showed that members in the family of networks represented by MDSXN inherit properties of both the de Bruijn graph and the hypercube to a varying degree. This allows network designers to construct networks with different cost and performance for different purposes. References [1] M. Sridhar and C. Raghavendra, Fault-tolerant networks based on the de Bruijn graph; IEEE Trans. on Comput., vol. 40, no. 10, Oct [2] C. Rose, Low mean internodal distance network topologies and simulated annealing, IEEE Trans. on Comm., vol. 40, no. 8, Aug. 1992, [3] Y. Saad and M. Schultz, Topological properties of hypercubes, IEEE Trans. on Comput., vol. 37, no. 7, Jul [4] E, Ganesan and D. Pradhan, The Hyper-deBroijn networks: Scalable versatile architecture, IEEE Trans. on Pa~ and Distr Sys., vol. 4, no. 9, Sep [5] B. Mukherjee, WDM-based local lightwave networks part II: Multihop systems~ IEEE Networks, [6] 1, Chlamtac, A, Ganz, and G, Karmi, Lightnets: Topologies for high-speed optical networks; IEEE Jour of Lighrwave Tech., vol. 11, no. 5/6, May/June [7] B. Mukherjee, S. Ramamurthy, D. Banerjee, and A. Mukherjee, Some principles for designing a wide-area optical network: Proc. of IEEE INFOCOM 94, [8] K. Sivarajan and R. Ramaswami, Lightwave networks based on de Brtrijn graphs: IEEE Trans. on Networking, vol. 2, no. 1, Feb transition in dimension O - - transition in dimension 1 Figure 8: A cycle of length 10 in the (4,2,[2 2]) MDSXN. [9] [10] R. Wilson, Introduction to Graph Theory. Academic Press, N. debruijn, A combinatorial problem, Koninklijke Netherlands: Academe Van Wetenschappen, vol. 49, no. 20, 1946.

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