OBLIVIOUS PERMUTATION ROUTING WITH THREE COLORS IN LOW-DIMENSIONAL BINARY DE BRUIJN NETWORKS Dobri Atanassov Batovski St. Gabriel Telecommunications Research Laboratory, Department of Telecommunications Science Faculty of Science and Technology, Assumption University Soi 24, Ram Khamhaeng Road, Hua Mak, Bang Kapi, Bangkok 10240 E-mail:
[email protected]
one packet per p planes per routing step on the average. This redundancy is provided to ensure the proper functioning of the oblivious scheme. As greater is the number of planes, as less efficient is the routing due to the increased transmission delays. It is preferable to obtain optimal SD graphs based on minimal number of planes. Computational experiments are used to obtain sample graphs where the computation ends with a number of planes that cannot be reduced any further.
ABSTRACT The three-color ability of the binary de Bruijn network is used to establish conflict-free oblivious packet routing within the all-port model. Tight multi-plane routing is implemented to obtain the optimal number of planes for permutation routing for a number of dimensions from two to six. Keywords: Binary de Bruijn network, graph coloring, oblivious routing.
2. THREE-COLOR CHANNEL ASSIGNMENT The directed binary d-dimensional de Bruijn network B(2, d) has N = 2d distinct nodes. Every node u is assigned a binary label (bd-1 ... b0), where bi = 0, 1. b0 is the least significant bit (LSB, right-most) and bd-1 is the most significant bit (MSB, left-most) of the label. A cyclic shift (rotation) of the d digits of u, denoted SH(u, b0), represents one of the nodes adjacent to u. The node (bd-1 ... b0) is adjacent to the 2 distinct nodes obtained with consequent alternation of the new MSB and having binary labels (bd-2 ... t0 l), denoted SH(u, l), where l is an arbitrary binary digit, l = 0, 1. The binary de Bruijn network has a 3-color ability and every node can be assigned a different color to have no neighbors with the same color. [2,6] This coloring property can be extended to routing by assigning a color (wavelength) to the output port of a node as shown in Fig. 1. Therefore, neighbors exchange data on three different wavelengths to reduce the effect of optical crosstalk. A single wavelength per node is most likely to be used in optical interconnection networks due to the simplified design and reduced cost of implementation. Data can be sent to the two neighbor nodes during different time slots. A joint packet frame containing several packets can be sent to both neighbors in this case.
1. INTRODUCTION The directed binary de Bruijn graphs [1] consist of nodes having two input and two output ports being able to receive/send packets from/to corresponding neighbor nodes. The nodes should be able to store packets of fixed length and forward the said packets using a prescribed routing algorithm. The static topology of the network allows one to use oblivious routing for which the path for a given source-destination pair is known in advance. It is assumed that clock synchronization is established among the nodes of the network and the time is slotted accordingly. A network cycle consisting of a prescribed number of time slots for the transmission of packet frames is provided. A packet frame includes a fixed number of packets from arbitrary sources to be routed to arbitrary destinations. Every frame is formed prior transmission based on the information obtained from an oblivious routing table. The routing table represents an oblivious source-destination (SD) graph where the packets in the frame to be transmitted during a single routing step (hop) within the network cycle are explicitly known. The small portion of the graph related to a particular node can be viewed separately in terms of incoming/departing packets per frame per routing step. When routing arbitrary permutations, some packet slots in the corresponding frames remain empty since there are N packets (N = 2d, where d is the dimension of the network) to be routed within r routing steps on p planes (p packets per frame). The total number of packet slots within a network cycle equal the product prN. There is
Figure 1. Three-color channel assignment of a node.
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2. OBLIVIOUS MULTI-PLANE ROUTING The concept of multi-plane routing is shown schematically in Fig. 2. It is assumed that the oblivious source-destination pairs can split into several conflict-free graphs to be routed independently on separate logical planes. [7]
Figure 3. B(2,1), B(2,2), B(2,3), and a sample 3-color channel assignment.
Figure 2. A schematic representation of multi-plane routing. For example, the oblivious paths for the two source-destination pairs 5→2 and 9→1 5 → 2 : 5 10 4 9 2 9 → 1 : 9 2 4 8 1, are conflicting during routing step 3 within the 3-color configuration since two packets at node 4 are scheduled for transmission to neighbors 8 and 9 during the same time slot. If no alternative conflict-free paths can be found on the same plane, one can search for unused edges on the other planes. The networks for a number of dimensions up to five are shown in Figs. 3-5. [3] The six-dimensional graph is shown in Fig. 6. [4] Higher dimensions can also be viewed using appropriate techniques. [5] A sample three-color channel assignment is demonstrated in Figs. 3-5. A folded de Bruijn network has been defined in [7] by replacing the two identity edges between the all-zero (0…0) and all-one (1…1) binary labels with an undirected edge. This reduces the network diameter of the undirected graphs and increases the number of alternative paths for oblivious routing. It should be noted that the reduced connectivity to nodes with labels (0101…), (1010…), (0…0), and (1…1) acts as a bottleneck for some paths during the routing process. Figure 7 shows a sample permutation graph in the B(2,4) network using 3 colors and 4 planes. The oblivious graphs are usually represented by their major paths, which cannot be derived from longer path sequences of labels. The labels are given in decimal format for convenience. Horizontal lines on the planes indicate that the packets stay put in the buffers of the nodes. One should keep in mind that in permutation (one source - one destination) routing the multiple paths that originate from same sources/destinations are conflict-free. Conflicts can be traced by observing non-horizontal path segments from different sources and destinations that merge together during a routing step.
Figure 4. B(2,4) and a sample 3-color channel assignment.
Figure 5. B(2,5) and a sample 3-color channel assignment.
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Figure 6. B(2,6). Plane Two
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Plane One 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 0
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15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 0
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Plane Three 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 0
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Figure 7. A sample permutation SD graph with four planes in the 3-color folded B(2,4) network. 4 → 0 : 4 8 0 4 → 8 : 4 8,
As an illustration of the concept of major and minor paths, the oblivious route
and a corresponding set of minor paths with respect to the destination address 14,
4 → 14 : 4 8 0 15 14,
indirectly represents a set of minor paths with respect to the initial source address 4,
8 → 14 : 8 0 15 14 0 → 14 : 0 15 14 15 → 14 : 15 14.
4 → 15 : 4 8 0 15
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Here the major path 4 → 14 uses the undirected edge 0↔15 of the folded network. The computational evaluation of the minimum number of planes can be performed for four basic network configurations. The standard one is based on four-color channel assignment with two colors for the input optical ports and two colors for the output ones. The folded configuration makes use of the additional undirected edge (0…0)↔(1…1). The 3-color node design introduced here is to be compared with the standard one. The results are presented in Table 1. As it should be expected, the number of planes increases rapidly with the dimension of the network. For a number of dimensions greater than six, the number of planes increases to impractical values. The routing on a single plane follows to the increase of the number of routing steps as shown in Table 2. A sample graph is shown in Fig. 8. The results in Tables 1 and 2 are obtained on the basis of exhaustive computations. Note that a significant decrease of the number of planes can be obtained with a minor increase of the number of routing steps for non-tight multi-plane routing.
3. CONCLUSION The choice of a binary network comes from technological constraints imposed on optical networks in terms of crosstalk. The 2x2 optical switching components are more likely to dominate for some time. The choice of alternative 2x1 optical components with the use of three colors for forwarding packets is an alternative for the design of optical interconnection networks. The number of planes/routing steps in the 3-color network configuration does not differ significantly from the standard one for low-dimensional binary de Bruijn networks. Thus, the use of nodes with a single output wavelength can be considered for possible hardware implementations in optical interconnection networks for reducing the costs of implementation and the power consumption as well.
4. ACKNOWLEDGMENTS This work is supported by the Assumption University under Research Grant RCF 3/2004. The Montfort Brothers of St. Gabriel, Assumption University, are gratefully appreciated.
Table 1. Number of planes, p, for tight permutation routing (r = d) for different dimensions and network configurations. Number of planes, p N Standard Standard, 3-Color Folded, Folded 3-color 2 1 1 2 1 3 2 1 3 2 4 4 2 6 4 5 8 7 11 9 6 13 12 20 18
5. REFERENCES [1] N.G. de Bruijn, “A Combinatorial Problem,” Proceedings of the Section of Science, Applied Mathematical Science, Koninklijke Nederlandse Akademie van Wetenschappen, ser. A, vol. 49, no. 2, pp. 758-764, 1946. [2] C. Berge, The Theory of Graphs and Its Applications, John Wiley Publishing, 1962.
Label
Table 2. Number of routing steps, r, for non-tight permutation routing on a single plane (p = 1) for different dimensions and network configurations. Number of routing steps, r n Standard Standard, 3-Color Folded, Folded 3-color 3 4 3 5 4 4 6 5 7 7 5 11 10
[3] S.W. Golomb, Shift Register Sequences, Holden-Day Incorporated, 1967. [4]
[5] D’H.J. Hunt, Constructing Higher-order de Bruijn Graphs, Thesis, Naval Postgraduate School, Monterey, California, U.S.A., June 2002.
Routing Plane
15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0
H. Taylor, “Drawing de Bruijn Graphs,” The Mathemagician and Pied Puzzler: A Collection in Tribute to Martin Gardner, E.R. Berlekamp and T. Rodgers (Eds.), A K Peters, Ltd., pp.197-198, 1999.
[6] J.-W. Mao, “The Coloring and Routing Problems on de Bruijn Interconnection Networks,” Ph.D. Dissertation, National Sun Yat-Sen University, Kaohsiung, Taiwan, R.O.C., July 2003. [7]
0
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3 4 Routing Step, r
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Figure 8. A conflict-free permutation SD graph with 7 routing steps in the 3-color B(2,4) network.
D.A. Batovski, “Packet Burst Switching and Multi-Wavelength Circuit Switching in the Binary de Bruijn Network,” Proceedings of the International Conference on Parallel and Distributed Processing Techniques and Applications (PDPTA’05), The 2005 International Multi-Conference in Computer Science & Computer Engineering, CSREA Press, June 27-30, 2005, Monte Carlo Resort, Las Vegas, Nevada, U.S.A., vol. I, pp. 105-111.
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