addressing and routing in the cube model of

ADDRESSING AND ROUTING IN THE CUBE MODEL OF MANHATTAN STREET NETWORK   AND TUG RUL DAYARy DAVUT TOKGOZ

Abstract. Manhattan Street Network (MS-Net) is a mesh-con gured, two-degree interconnected network. Reduced binary addressing scheme (reduced-BAS) is a simple addressing methodology that provides a convenient solution for inserting new nodes and deleting existing ones in a mesh network. When inserting and deleting nodes, the addresses of existing nodes do not change. Hence, reduced-BAS overcomes addressing diculties present in MS-Net. A third dimension is added to the conventional MS-Net and reduced-BAS is presented for this new topology. The extra dimension of MS-Net provides a smaller average number of hops between a source-destination pair than the conventional MS-Net with approximately the same number of nodes and therefore improves eciency. The new routing rules can be easily implemented in hardware or in software. Key Words. Manhattan Street Network, reduced binary addressing scheme, cube model

1. Introduction. In 1985, a metropolitan area network (MAN) architecture and a

routing protocol called Manhattan Street Network (MS-Net) [1] was proposed by AT&T Bell Laboratories. This network is formed of N  N nodes connected in a toroidal topology. We can de ne MS-Net as a regular, two-degree interconnected, mesh-con gured network with unidirectional links. The nodes are connected on a toroidal surface, with adjacent rows and columns carrying data packets in opposite directions similar to the streets and avenues in Manhattan. Nodes are addressed sequentially. Because of its toroidal surface, there are no corners (see Figure 1) [1, p. 842]. Each node has two incoming links and two outgoing links (see Figure 2) [1, p. 842]. The arriving packets are latched at buers D and designated packets are put into the buers of outgoing links. There are also incoming and outgoing links attached to the local source. In each node  packets are delayed at the inputs so that they arrive at the switch simultaneously,  packets destined to the node are removed,  each outgoing packet from the local source is inserted to an empty slot,  an output link is selected at the switch for each outgoing packet. The operation of the network is based on two assumptions [2, p. 1660]: Assumption 1: The arrival of a packet on an incoming link is independent of an arrival on the other incoming link. Assumption 2: The choice of an output link for a packet is independent of the choices for other packets. Two addressing schemes, namely integer and fractional addressing, had been proposed for mesh networks. The former causes a modi cation in the network when new nodes are inserted. The latter requires more computing time when routing a packet. Due to these problems, a new addressing scheme called reduced binary addressing scheme (reduced-BAS) [6] has been proposed for mesh networks. Reduced-BAS provides exibil Application Software Development Department, NETAS, Cinnah Caddesi 10/A, Kavakldere 06690, . Ankara, Turkey ([email protected]). y Department of Computer Engineering and Information Science, Bilkent University, Bilkent 06533, Ankara, Turkey ([email protected]).

Figure 1.

4  4 MS-Net with reduced-BAS.

Figure 2.

The structure of a node.

ity when inserting new nodes to the network. Another important bene t of this scheme is the reduced computation time of routing. Moreover, the routing rules in reducedBAS can be implemented using very simple hardware. Several comparative performance studies of MS-Net may be found in [2, 3, 4, 5, 7, 8]. The proposed cube model of MS-Net is three-degree interconnected and has a regular structure just like the two-degree MS-Net. Nodes in this model are connected to unidirectional links along x; y, and z dimensions which carry packets in orthogonal directions (see Figure 3). Each node in this topology has three incoming links and three outgoing links (see Figure 4). Nodes in the cube model of MS-Net must also be addressed sequentially. If nodes are not addressed sequentially, then packets may follow longer than necessary paths. The next section presents reduced-BAS for the cube model of MS-Net. The routing rules of the addressing scheme appear in section 3 and a hardware implementation is given in section 4. We end the paper with concluding remarks.

2. Reduced binary addressing scheme in MS-Net. Two addressing schemes

for MS-Net are discussed in [6]. The rst one is the integer addressing scheme that assigns each node a two dimensional absolute address with an x dimension component and a y dimension component. In the cube model of MS-Net, the same scheme assigns a three dimensional absolute address to each node (e.g., (0, 1001, 1011)). The problem with this addressing scheme is that the addresses of existing nodes must be modi ed when

Figure 3.

4  4  4 MS-Net with reduced-BAS.

new nodes are inserted to the network. The second addressing scheme uses fractional addresses which are formed of real numbers (e.g., (10.1001, 0.001, 10.1)). It allows any number of nodes to be inserted at any position along x; y; and z dimensions of the cube model MS-Net without changing the addresses of existing nodes. The drawback of this scheme is that the routing decision is more time consuming. A new addressing scheme called reduced-BAS attempts at solving the problems associated with the earlier addressing schemes proposed for MS-Net. Inserting new nodes using reduced-BAS is simple since there is no need to change existing node addresses. Reduced-BAS also improves the time for routing. This issue is addressed in section 3. Figure 5(a) shows a three-bit binary number axis representing 8 node addresses labeled from `000' to `111'. If a new node is inserted to the axis, all the addresses of existing nodes have to be modi ed. While doing this, an extra bit is appended to existing addresses using one of two approaches. The rst approach is to append a bit to the beginning of the addresses. If we append a `1' (`0') to the beginning of the existing addresses, then 8 new nodes can be inserted before (after) existing nodes on the axis with rst bit `0' (`1'). In the second approach, the fourth bit is appended to the end of the addresses. Hence, only a single new node can be inserted between two nodes. However, this approach will change all existing nodes into even or odd numbered nodes. So, the property of adjacent links carrying data in opposite directions cannot be maintained easily when new nodes are inserted. In Figure 5(a), the reduced-BAS axis appears beside the three-bit binary number axis with 8 nodes. To obtain reduced binary numbers, the least signi cant bit in the corresponding binary number is eliminated recursively as long as it is equal to the next least signi cant bit (e.g., `011' ) `01', `111' ) `1'). This new

Figure 4.

The structure of a node in the cube model of MS-Net.

Figure 5.

An illustrative example concerning reduced-BAS.

addressing scheme provides the possibility of inserting new nodes to the reduced binary axis without modifying existing addresses. This is shown in Figure 5(b). In this gure, 8 new nodes are inserted to the reduced binary axis. As a result, a 16 node reduced binary axis is obtained. Nodes `0001' and `0010' are inserted between `0' and `001'. The inserted nodes are denoted by solid circles. The following de nitions help us better understand the merits of reduced-BAS [6, p. 27]. Definition 1. (Reduction Rule) When the least signi cant n bits of an m-bitaddress are identical, reduced-BAS collapses the n bits to a single bit, thus shortening the bit string to m n + 1 bits. Definition 2. (Extension Rule) Given two address strings of dierent lengths, the shorter address string must be extended by duplicating its least signi cant bit to the length of the longer address string before an operation on the strings can be performed. Definition 3. Two nodes are called tightly connected if there is no address space between them. The rst rule de nes reduced-BAS. The distance between two nodes may be computed by subtracting one of the addresses from the other. For reduced binary numbers, the extension rule must be applied prior to the subtraction. The shorter address should

Figure 6.

Inserting nodes to an existing network (irregular MS-Net).

be extended to the length of the longer one. The subtraction is similar to the one for integer addressing, however the result must also contain sign bits. By using the above rules we can de ne the sequence of reduced binary addresses. For example, addresses `01' and `10' are in between `0' and `1', addresses `101' and `110' are in between `10' and `1', and addresses `001' and `010' are in between `0' and `01'. In this way the address space can be extended to any size. We should also remark that the reduced binary axis is piecewise-continuous meaning the axis is neither continuous nor discrete. New nodes can be inserted only in the segments that are continuous. The third de nition is introduced for this purpose. These properties lead to the following theorem [6, p. 27]. Theorem 1. Let U and V be two reduced-BAS addresses. U and V are tightly connected if U = W  01 and V = W  10, where W is any string and `' is the concatenation operator. For example, consider the pairs of nodes f`01',`10'g and f`001',`010'g in Figure 5. Both pairs are tightly connected. The theorem is about the piecewise-continuity property of the reduced binary axis and implies the existence of tightly connected pairs. In Figure 6, we show how one can insert 4 new nodes to a two dimensional MS-Net. Figure 3 shows the cube model of MS-Net (i.e., three dimensions) with reduced-BAS and Figure 7 has its blown up version. The main feature of this addressing scheme is its provision for easy node insertion and deletion. An example of node insertion to this network is given in Figure 8. In the gure, 8 new nodes are inserted to a three dimensional MS-Net. When inserting new nodes, the basic structure of MS-Net should be preserved. Hence, nodes can be inserted with increments of 2i  2j  2k, where i; j , and k are integers. This means 2i (x dimension) nodes plus 2j (y dimension) nodes plus 2k (z dimension) nodes must be inserted simultaneously. The reason for this is to ensure adjacent link ows in opposite directions along each dimension. If fewer nodes are to be inserted, then dummy nodes can be used to preserve structure. The next section presents routing rules that may be implemented in hardware or software for the cube model of MS-Net.

3. Routing in the cube model of MS-Net. In any kind of packet switched metropolitan or wide area network, several hundred thousand routing decisions are made

Figure 7.

Part of three dimensional MS-Net.

at each node. Therefore, simplicity and speed of routing is quite important. When a packet is routed through a node, one of the output links is selected based on prede ned rules taking into account the current node and the destination node. The aim of a routing rule is to minimize the number of hops traveled by packets in the network. There are many proposed rules for routing, some good and some not so good. For a given packet in a node, the output link selected for routing is called the preferred link. Reduced-BAS provides a very simple mechanism to determine the preferred link. The following de nition is useful in constructing the routing rules [6, p. 28]. Definition 4. Assume a packet arrives at node C destined to node D in the cube model of MS-Net. Let (Xc; Y c; Zc) and (Xd; Y d; Zd) be current and destination node addresses, respectively. Then the relative (routing) address RA of the packet is de ned as (Xr; Y r; Zr) := (Xd Xc; Y d Y c; Zd Zc). Routing in the three dimensional MS-Net is similar to that of the conventional model. There are eight types of nodes in this model (see Figure 7). Assuming the nodes of interest are addressed as (Xc; Y c; Zc),  type 1 nodes have outgoing links to nodes (Xc + 1; Y c; Zc), (Xc; Y c + 1; Zc), and (Xc; Y c; Zc + 1),  type 2 nodes have outgoing links to nodes (Xc 1; Y c; Zc), (Xc; Y c + 1; Zc), and (Xc; Y c; Zc + 1),  type 3 nodes have outgoing links to nodes (Xc 1; Y c; Zc), (Xc; Y c 1; Zc), and (Xc; Y c; Zc + 1),  type 4 nodes have outgoing links to nodes (Xc + 1; Y c; Zc), (Xc; Y c 1; Zc), and (Xc; Y c; Zc + 1),  type 5 nodes have outgoing links to nodes (Xc + 1; Y c; Zc), (Xc; Y c + 1; Zc), and (Xc; Y c; Zc 1),  type 6 nodes have outgoing links to nodes (Xc 1; Y c; Zc), (Xc; Y c + 1; Zc), and (Xc; Y c; Zc 1),  type 7 nodes have outgoing links to nodes (Xc 1; Y c; Zc), (Xc; Y c 1; Zc), and (Xc; Y c; Zc 1),  type 8 nodes have outgoing links to nodes (Xc + 1; Y c; Zc), (Xc; Y c 1; Zc), and (Xc; Y c; Zc 1). Nodes may be distinguished by inspecting the least signi cant bits of their address com-

Figure 8.

Inserting new nodes to an existing network (irregular three dimensional MS-Net).

ponents. Note that there are 8 permutations of the three least signi cant bits (i.e., 2 permutations in each of the three dimensions). Nodes type 2 to 8 can be translated to the address space of the rst type of nodes by negating (i.e., taking the two's complement of) the appropriate relative address components. Hence, we can apply the same link selection rules to nodes type 2 to 8 as the rst type of nodes after a simple modi cation of the relative address (see [6, pp. 28{29]). Due to the toroidal structure of MS-Net, any node can be perceived as being at the center of the network. First the relative address of the packet to be routed is computed from De nition 4. There are eight regions into which relative addresses fall (see Figure 9) and they are given by  R1 := f(Xr; Y r; Zr) j 0 < Xr  1; 0 < Y r  1; 0 < Zr  1g,  R2 := f(Xr; Y r; Zr) j 1  Xr  0; 0 < Y r  1; 0 < Zr  1g,  R3 := f(Xr; Y r; Zr) j 1  Xr  0; 1  Y r  0; 0 < Zr  1g,  R4 := f(Xr; Y r; Zr) j 0 < Xr  1; 1  Y r  0; 0 < Zr  1g,  R5 := f(Xr; Y r; Zr) j 0 < Xr  1; 0 < Y r  1; 1  Zr  0g,  R6 := f(Xr; Y r; Zr) j 1  Xr  0; 0 < Y r  1; 1  Zr  0g,  R7 := f(Xr; Y r; Zr) j 1  Xr  0; 1  Y r  0; 1  Zr  0g,  R8 := f(Xr; Y r; Zr) j 0 < Xr  1; 1  Y r  0; 1  Zr  0g. According to the information above, the following rules can be applied to the rst type of nodes. Rule 1. If the destination node is in R1, then Lx(Xc + 1; Y c; Zc) or Ly (Xc; Y c + 1; Zc) or Lz(Xc; Y c; Zc + 1) can be selected. If incoming direction is along Lx, then Lx is selected, else if incoming direction is along Ly then Ly is selected, else if incoming direction is along Lz, then Lz is selected. Otherwise the packet comes from local source and the following conditions are tested:  If (Xr < Y r) and (Xr < Zr), then Lx(Xc + 1; Y c; Zc) is selected,

 else if (Y r < Xr) and (Y r < Zr), then Ly(Xc; Y c + 1; Zc) is selected,  else if (Zr < Xr) and (Zr < Y r), then Lz(Xc; Y c; Zc + 1) is selected,  else if (Xr = Y r) and (Xr < Zr), then Lx(Xc +1; Y c; Zc) or Ly(Xc; Y c +1; Zc)

can be selected (there are no preferred links and one of them is chosen randomly),  else if (Xr = Zr) and (Xr < Y r), then Lx(Xc +1; Y c; Zc) or Lz(Xc; Y c; Zc +1) can be selected (there are no preferred links and one of them is chosen randomly),  else if (Y r = Zr) and (Y r < Xr), then Ly(Xc; Y c +1; Zc) or Lz(Xc; Y c; Zc +1) can be selected (there are no preferred links and one of them is chosen randomly),  else if Xr = Y r = Zr, then Lx(Xc + 1; Y c; Zc) or Ly(Xc; Y c + 1; Zc) or Lz(Xc; Y c; Zc + 1) can be selected (there are no preferred links and one of them is chosen randomly). Rule 2. If the destination node is in R2, then Ly (Xc; Y c +1; Zc) or Lz (Xc; Y c; Zc + 1) can be selected. If incoming direction is along Ly, then Ly is selected, else if incoming direction is along Lz, then Lz is selected. Otherwise the packet comes from local source and the following conditions are tested:  If Y r < Zr, then Ly(Xc; Y c + 1; Zc) is selected,  else if Zr < Y r, then Lz(Xc; Y c; Zc + 1) is selected,  else if Zr = Y r, then Ly(Xc; Y c + 1; Zc) or Lz(Xc; Y c; Zc + 1) can be selected (there are no preferred links and one of them is chosen randomly). Rule 3. If the destination node is in R3, then Lz (Xc; Y c; Zc + 1) is the preferred link. Rule 4. If the destination node is in R4, then Lx(Xc +1; Y c; Zc) or Lz (Xc; Y c; Zc + 1) can be selected. If incoming direction is along Lx, then Lx is selected, else if incoming direction is along Lz, then Lz is selected. Otherwise the packet comes from local source and the following conditions are tested:  If Xr < Zr, then Lx(Xc + 1; Y c; Zc) is selected,  else if Zr < Xr, then Lz(Xc; Y c; Zc + 1) is selected,  else if Zr = Xr, then Lx(Xc + 1; Y c; Zc) or Lz(Xc; Y c; Zc + 1) can be selected (there are no preferred links and one of them is chosen randomly). Rule 5. If the destination node is in R5, then Lx(Xc + 1; Y c; Zc) or Ly (Xc; Y c + 1; Zc) can be selected. If incoming direction is along Lx, then Lx is selected, else if

incoming direction is along Ly, then Ly is selected. Otherwise the packet comes from local source and the following conditions are tested:  If Xr < Y r, then Lx(Xc + 1; Y c; Zc) is selected,  else if Y r < Xr, then Ly(Xc; Y c + 1; Zc) is selected,  else if Xr = Y r, then Lx(Xc + 1; Y c; Zc) or Ly(Xc; Y c + 1; Zc) can be selected (there are no preferred links and one of them is chosen randomly). Rule 6. If the destination node is in R6, then Ly (Xc; Y c + 1; Zc) is the preferred link. Rule 7. If the destination node is in R7, then Lx(Xc + 1; Y c; Zc) or Ly (Xc; Y c + 1; Zc) or Lz(Xc; Y c; Zc + 1) can be selected. The following conditions are tested:  If (jXrj > jY rj) and (jXrj > jZrj), then Lx(Xc + 1; Y c; Zc) is selected,  else if (jY rj > jXrj) and (jY rj > jZrj), then Ly(Xc; Y c + 1; Zc) is selected,  else if (jZrj > jXrj) and (jZrj > jY rj), then Lz(Xc; Y c; Zc + 1) is selected,  else if (jXrj = jY rj) and (jXrj > jZrj), then Lx(Xc + 1; Y c; Zc) or Ly(Xc; Y c + 1; Zc) can be selected (there are no preferred links and one of them is chosen randomly),  else if (jXrj = jZrj) and (jXrj > jY rj), then Lx(Xc+1; Y c; Zc) or Lz(Xc; Y c; Zc +1) can be selected (there are no preferred links and one of them is chosen randomly),  else if (jY rj = jZrj) and (jY rj > jXrj), then Ly(Xc; Y c+1; Zc) or Lz(Xc; Y c; Zc +1) can be selected (there are no preferred links and one of them is chosen randomly),  else if jXrj = jY rj = jZrj, then Lx(Xc + 1; Y c; Zc) or Ly(Xc; Y c + 1; Zc) or Lz(Xc; Y c; Zc + 1) can be selected (there are no preferred links and one of them is chosen randomly). Rule 8. If the destination node is in R8, then Lx(Xc + 1; Y c; Zc) is the preferred link. 4. Hardware implementation of routing rules. Figure 10 shows a hardware implementation of the routing rules for a three dimensional MS-Net with reduced-BAS. The time needed for the routing decision depends only on gate delays. Thus, due to the pure combinational nature of the circuit, decision time is negligible. First, the destination address is subtracted from the current node address after applying the extension rule in De nition 2. Then 2's complement of the appropriate relative address component is taken based on the least signi cant bits of the current node address components. taken. This is done to convert the address space of nodes type 2 to 8 to the address space of type 1 nodes. The output goes into the comparator which selects the preferred link using the link selector. If there is more than one candidate for the preferred link, the comparator selects one randomly. When routing a packet to the destination, the current node selects the preferred link without knowledge of the addresses of adjacent nodes. The regular structure of the MS-Net will not be preserved if new nodes are inserted, existing nodes are deleted or they fail. Hence, the routing rules discussed do not work properly in the irregular parts of the network in some cases. For example, in Figure 8, assume a packet is destined from node A to node C . After computing the relative address, node C turns out to be in R6 with respect to Figure 9. According to Rule 6, link Ly is selected and the packet is transmitted to node B . However, this link is local and the packet moves in the irregular

Figure 10.

Simple hardware for routing.

part of the network. Node B recognizes that this packet is not destined to itself and routes it. Based on the relative address, the packet is in R5 and Rule 5 selects link Ly. Consequently, the packet returns to node A. The packet of interest is stuck in a loop. Similarly, a packet from node B destined to node E rst goes to node D but then returns to node B according to the routing rule just mentioned. We propose two approaches to prevent a packet from getting stuck in a loop. The rst one is to use random routing. This is the simplest solution and can prevent packets from looping. However, this approach will most likely cause a packet to follow a longer source destination path. Besides, some of the links in the irregular part may be used more than once. A remedy is to employ random routing only in the irregular parts of the network. However, this requires dierent hardware for regular and irregular parts. The second approach is to store irregular domain information and addresses of adjacent nodes in irregular nodes. With the help of this information, a packet can leave the irregular parts of the network using alternative paths. The proposed approach, which guarantees that each link in the source destination path is taken once, requires additional software. Here we propose a hardware solution (see Figure 11) in which the hardware of all nodes are identical and looping in the irregular part of the network is avoided. A circuit called the link allower circuit is added to the hardware of Figure 10. This circuit includes a 3  8 decoder and some switches. The link allower circuit determines the links allowable for each region. The comparator can then select a link from the allowable link set. For example, if the destination address is in R2, then the link allower circuit sends an output to the comparator in which Ay and Az are on and Ax is o. This means that the comparator can select either Ly or Lz as the preferred link. In fact, the comparator selects one of these links without the link allower circuit when the destination address is in R2. Then how must one use the link allower circuit? In the example concerning the problem with routing in irregular parts of the network, node B does not have to prefer link Ly when the destination node is in R5. If node B selects link Lx, then the packet

Figure 11.

Another solution for solving the routing problem with hardware.

will not get stuck in a loop. However, according to Rule 5, it is a possibility for link Ly to be selected. If we turn o the switch to Ly (of R5) in the link allower circuit, the comparator will have no choice but to select link Lx. After pinpointing the irregularities in routing for node B , we can reorganize the switches for this particular node as in Figure 12. In this way, a packet avoids getting stuck in a loop and passes through each node along the source-destination path once. Other irregular nodes can be treated in a similar manner. 5. Merits of the cube model of MS-Net. When an extra dimension is introduced to the conventional MS-Net, the number of incoming and outgoing links both increase by one. Assuming hardware cost is a linear function of the number of links and buers, the cost is fty percent higher in the cube model of MS-Net. However, the improvement in the average distance between source-destination pairs in the cube model is likely to justify the extra hardware cost. Theorem 2. Assuming that source-destination pairs for packets are uniformly distributed across a d dimensional MS-Net with npnodes, the average distance between one such source-destination pair is proportional to d n. Proof. Due to the way in which routing rules of section 3 are implemented, a packet stops moving along a given dimension when the address component of RA along that dimension becomes 0. Since packets move one step closer to the destination along one of the dimensions at each routing decision, the statement of the theorem is proved. 2 In a two dimensional N N MS-Net, the average distance between a source-destination pair along the x dimension is PN 1 i N 1 i=1 = N 2 : The average distance along the y dimension is the same. As a result, the total average

Figure 12.

Hardware solution, as in Figure 11, for node B of Figure 8.

distance is N p1. Since n = N 2, the average distance between a source-destination pair is obtained as n 1. Similarly, in a three dimensional M  M  M MS-Net, the average distance between a source-destination pair is 1:5(M 1). Since n = M 3, the average distance is calculated p to be 1:5( 3 n 1). For a fty percent improvement in the average distance of a sourcedestination pair in a three dimensional MS-Net (compared to that of a two dimensional MS-Net),

pn 1 p 1:5( n 1)  2 2

3

must be satis ed. When n is greater than 416 (i.e., for a network of at least 416 nodes) such an improvement in average distance may be achieved. 6. Conclusion. Routing in MS-Net is more complicated than in linear topology networks. Inserting and deleting nodes in MS-Net using conventional addressing schemes are dicult since these changes require existing node addresses to be modi ed. In this paper, the cube model of MS-Net is presented. This model provides shorter paths between source-destination pairs than those provided by the conventional two dimensional model. Reduced binary addressing scheme is modi ed so that node insertion and deletion in the cube model are accomplished easily. In addition to these, ecient and fast routing rules that can be implemented in hardware or software are given. A hardware solution to routing with reduced-BAS is presented. Finally, a small modi cation is made in hardware to solve the problems associated with looping. In summary, reduced-BAS can be used with two or higher degree connected networks and extra address dimensions can be introduced to MS-Net to improve eciency at the expense of increased hardware cost.

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