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tolerant routing in metacube. The remainder of this paper isorganized as follows. Section 2 introduces the MC networkand its topological properties. Section 3 gives the routing algorithmin metacube. Section 4 introduces a multi-channelcube structure which is used for fault-tolerant routing in themetacube. Sections 5 gives the fault-tolerant routing algorithm.Section 6 concludes the paper and presents somefuture research directions.2. PreliminariesThis section formally introduces the MC network, itstopological properties and some related notation. The MCnetwork is motivated by the dual-cube network proposed byLi and Peng [6] [7] that mitigates the port limitation problemin the hypercube network so that the number of nodesin the network is much larger than that of the hypercubewith a fixed amount of link per node. The MC network includesthe dual-cube as a special case. An MC network hasa 2-level cube structure: high-level cubes represented bythe leftmost k bits of the binary address of the node whichcontains m2 k + k bits (these k bits serve as a class indicator),and low-level cubes, called clusters that form the basiccomponents in the network, represented by the m bits of theremain m2 k bits, which occupy the different portions in them2 k bits for different classes.More specifically, there are two parameters in an MCnetwork, k and m. An MC(k,m) contains h = 2 k classes.Each class contains 2 m(h−1) clusters, and each cluster contains2 m nodes. Therefore, an MC(k,m) uses mh + k binarybits to identify a node and the total number of nodesis 2 n where n = mh + k. The value of k affects stronglythe growth rate of the size of the network. An MC(1,m)containing 2 2m+1 nodes is called a dual-cube. Similarly, anMC(2,m), an MC(3,m) and an MC(4,m) containing 2 4m+2nodes, 2 8m+3 nodes and 2 16m+4 nodes are called quad-cube,oct-cube and hex-cube, respectively. Since an MC(3,3) contains2 27 nodes, the oct-cube is sufficient to construct practicallyparallel computers of very large size. The hex-cubeis of theoretical interest only. Note that an MC(0,m) is ahypercube.A node in an MC(k,m) can be uniquely identified bya (mh + k)-bit binary number. The leftmost k-bit binarynumber defines a class of the node (class_id). There areh classes. In each class, there are 2 mh nodes and each nodeis represented by a mh-bit binary number. 2 m nodes of thesame class form a cluster. Therefore, there are 2 m(h−1) clustersin each class. An m-bit binary number, located in aspecial portion of the mh-bit (will be explained in the nextparagraph) identifies a node within the cluster (node_id).Therefore, the (mh + k)-bit node address in an MC(k,m) isdivided into three parts: a k-bit class_id, an m(h − 1)-bitcluster_id and an m-bit node_id.In the following discussion, we use u = (c u ,M u [h −1],...,M u [1],M u [0]) to denote the address of node u,where c u is a k-bit binary number and M u [i], 0 ≤ i ≤h − 1 are m-bit binary numbers. Let class_id(u) =c u , node_id(u) = M u [c u ] × 2 c uand cluster_id(u) =∑ h−1i=0 M u[i] × 2 i − node_id(u). The mh-bit numbernode_id(u) + cluster_id(u) is a unique identifier of nodeu in class c u . For example, u = 0100111000 in an MC(2,2)is denoted as node 56 of class 1 and node set (48,52,56,60)in class 1 forms a cluster with cluster_id = 48.The links of an MC(k,m) is constructed in the followingmanner. The m-bit field M[c] in the address of a nodeof class c forms a low-level m-cube with m links, namelycube-edges. These low-level m-cubes are called clusters.A cluster containing node u is denoted as C u . The linksthat connect nodes among clusters are called cross-edgesand are defined as following. For any two nodes whose addressesdiffer only in a bit position in the class field, thereis a cross-edge connecting these two nodes. That is, the k-bit field c forms a high-level k-cube which connects thosenodes whose addresses except class field are the same.00000000100100001010080000100011010010101110000110001001011010021010011100101101111010001110011001111011131010111101101111111100100001100110001110class 0 class 1class 0Figure 1. A metacube MC(1,2)The addresses of two nodes connected by a cross-edgediffer only on one bit position within the k-bit class field andthere is no direct connection among the clusters of the sameclass. Therefore, a node in an MC(k,m) has m + k links: mlinks construct an m-cube cluster and k links construct a k-cube. For example, the neighbors in the cluster of the nodewith address (01,111,101,110,000) in an MC(2,3) haveaddresses (01,111,101,111,000), (01,111,101,100,000) and(01,111,101,010,000). The underlined bits are those thatdiffer from the corresponding bits in the address of the referencednode. The two neighbors in the high-level cube are(00,111,101,110,000) and (11,111,101,110,000).Fig. 1 shows the structure of an MC(1,2), where the clusteris a 2-cube and there are two classes. Each node has a41200101001110110101111344
0 14 50 41 52 36 78 129 138 912 132 63 710 1114 1510 1411 15252 253class 0 class 1254 255243 247251 2550 161 170 641 6532 4833 49128 192129 1932 183 192 663 6734 5035 51130 194131 195207 223class 2 class 3239 25563 127191 255Figure 2. A metacube MC(2,2)cross-edge attached to a node of the different class. The binarynumber shown in the center of a cluster is cluster_id.Fig. 2 shows the structure of an MC(2,2), where the clustersin the same square are of the same class. The decimalnumbers are node_id + cluster_id. In Fig. 2, there are2 2(22 −1) = 64 clusters in each square and each cluster is a2-cube. The figure shows only 4 high-level cubes, each ofwhich contains a distinct node in the cluster 0 of the class 0.The ratio of the total number of links in the hypercubeto the total number of links in the MC network is equal ton/(m + k), where n = m2 k + k. For example, for k = 2 andm = 3 (n = 14), each of the two networks contains 16384nodes; the hypercube contains 16384 × 14/2 = 114688links and the MC network contains 16384 × (3 + 2)/2 =40960 links. The reduction in the total number of links forthis example is 73728 links or about 64%.3. Point-to-point routing in metacubeThe problem of finding a path from a source node s toa destination node t, and forwarding messages along thepath is known as the point-to-point routing problem. It isthe basic problem for any interconnection network. In thissection, we describe briefly the point-to-point routing algorithmin metacube [8]. This algorithm is the building blockfor the proposed fault-tolerant routing algorithm.We adopt the following notation. In the metacubeMC(k,m), each node has m + k neighbors. Let s (i) , 0 ≤i ≤ k − 1, be the ith dimensional neighbor of node s withinthe k-cube, that is, the addresses of s and s (i) differ in theith bit position (the rightmost bit is the 0th bit) in the classfield c. Let s (i+k) , 0 ≤ i ≤ m − 1, be the ith dimensionalneighbor of node s in the m-cube, that is, the addresses of s345
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Page 7 and 8: To show the k paths for each node p

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