Fault-tolerant Routing in Metacube - CiteSeerX

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and s (i+k) differ in the ith bit position in the field M[c]. Lets (i, j) = (s (i) ) ( j) for 0 ≤ i, j ≤ m + k − 1. We use (u → v) todenote a path from node u to node v. If the length of a path(u → v) is 1 (through a single edge), the path is denoted as(u : v), and the edge is denoted as (u,v).In a graph G = (V,E) where V is the set of all vertices(nodes) and E is the set of all edges in G, let P ′ = (v 0 →v h−1 ) = (v 0 : v 1 : ... : v h−1 ) be a path from node v 0 to nodev h−1 , where v i ∈ V for 0 ≤ i ≤ h−1 and edge (v j−1 ,v j ) ∈ Efor 1 ≤ j ≤ h − 1. We say P ′ is a hamiltonian path if (1) P ′contains every node in V and (2) nodes v i (0 ≤ i ≤ h−1) areall distinct. Let P = (v 0 → v h ) = (P ′ : v h ), where v h = v i , fori = 0,1,..., or h − 2. If v h = v 0 , then P becomes a hamiltoniancycle 1 ; otherwise, we call P an extended-hamiltonianpath. The length of a hamiltonian path in a k-cube is2 k − 1; the length of a hamiltonian cycle or an extendedhamiltonianpath is 2 k . Let a weak-hamiltonian path bea hamiltonian path, a hamiltonian cycle, or an extendedhamiltonianpath. It was shown in [8] that given any twonodes s and t in an n-cube, there exists a weak-hamiltonianpath from s to t.For each node u in the k-cube, let next(u) be the nodenext to u in the weak-hamiltonian path from c s to c t . Let thenode addresses of s and t be (c s ,M s [h−1],...,M s [1],M s [0])and (c t ,M t [h − 1],...,M t [1],M t [0]), respectively. For therouting within an m-cube of class c, we can follow the ascendingrouting strategy, by which the least significant nonzerobit of (M s [c] ⊕ M t [c]) is chosen as the first dimensionfor routing, and so on. The routing algorithm in an MC(k,m)is given below. The Loop will terminate when the breakis executed. Notice that the details of routing in the m-cubeis omitted in the algorithm.Algorithm 1 (One_To_One_Routing(m,k,s,t))begin /* build a P = (s → t) in MC(k,m) */u = c s ; v = s; P = v;loop alwaysw = (u,M v [h − 1],...,M v [u + 1],M t [u],M v [u − 1],...,M v [0]);if (w ≠ v) P = (P → w);if (w == t) break;v = w;w = (next(u),M v [h − 1],...,M v [u + 1],M v [u],M v [u − 1],...,M v [0]);P = (P : w);u = next(u);endloopend.In the fault-tolerant routing we discuss late, we need directedhamiltonian cycles. Therefore, we introduce a pa-1 A hamiltonian cycle is defined as a path through a graph which startsand ends at the same vertex and includes every other vertex exactly once.rameter, next(s), to assign the direction of a hamiltoniancycle. For example, in Example 1, if we let next(s) = 10,then the hamiltonian cycle for (C s → C t ) in the high-level2-cube will be (00 : 10 : 11 : 01 : 00).Let H i (s,t), 0 ≤ i ≤ h − 1, be the Hamming distancebetween s and t in M[i], i.e. the number of bits with distinctvalues in M s [i] and M t [i]. From the algorithm, thelongest length of the routing path is 2 k + H h (s,t), whereH h (s,t) = ∑ h−1i=0 H i(s,t). This formula gives an upper boundto d(s,t), the distance between s and t in an MC(k,m). LetH(s,t) be the Hamming distance between s and t. Clearly,we have H(s,t) ≤ d(s,t) ≤ H h (s,t) + 2 k . Because H(s,t) =H h (s,t) + H k (s,t), where H k (s,t) is the Hamming distancebetween s and t in c field, we have H(s,t) ≤ d(s,t) ≤H(s,t) − H k (s,t) + 2 k . The longest path in an MC(k,m) isfrom s = 0···0 to t, where c t = 0···0 and M t [i] = 1···1 forall i, 0 ≤ i ≤ h − 1. The length of this path is 2 k (m + 1). Itis easy to see that this path is the shortest path for connectings and t. Therefore, it is the diameter of an MC(k,m).Since the average distance in each cluster is m/2, the averagedistance between any two nodes in an MC(k,m) isat most (m/2)2 k + 2 k = (n − k)/2 + 2 k , where n = m2 k + k(in the case of hamiltonian path, it is (n − k)/2 + 2 k − 1).Notice that it is possible to have a routing algorithm in anMC(k,m) which bypasses the class c if M s [c] = M t [c]. Insuch a case, the length of the routing path for some s and tmight be shorter than that produced by the algorithm above.4. Multi-channel cubeIn this section, we will describe a structure in themetacube, called multi-channel cube. This structure is usefulfor designing algorithms in MC(k,m) based on hypercubealgorithms. We will use this structure for solving faulttolerantrouting problem in the metacube.Let C u and C v be two distinct clusters in an MC(k,m),h = 2 k . A multi-channel cube is defined as C u ∪ C v ∪ E,where E is a set of k2 m disjoint paths connecting C u andC v , k paths for each node pair (u i ,v i ), u i ∈ C u , v i ∈ C v , fori = 0,1,...,2 m − 1. In the other word, E = ∪i, j(u i → v i ) j ,0 ≤ i ≤ 2 m −1 and 0 ≤ j ≤ k −1, and (u i1 → v i1 ) j1 ∩(u i2 →v i2 ) j2 = /0 if i 1 ≠ i 2 or j 1 ≠ j 2 .Figure 3 shows an example of a multi-channel cube. Thedotted line denotes a path, not an edge. In the case of atraditional n-cube, there is an edge (u,v) connecting twonodes u and v for u ∈ 0-subcube and v ∈ 1-subcube. Insteadof an edge, the two nodes of a node pair in a multi-channelcube are connected by k disjoint paths.The major problem in constructing the k disjoint paths isthat the path (u i → v i ) j , 0 ≤ i ≤ 2 m − 1 and 0 ≤ j ≤ k − 1,will just advance through cross-edges when M ui [l] = M vi [l]for some l, 0 ≤ l ≤ 2 k − 1. This will cause two paths346
k pathsu0v 0C uC vFigure 3. A multi-channel cubealong the distinct dimensions in the k-cube to intersect atsome vertex. For example, in an MC(2,2), the paths from0000000000 to 1111000000 along dimensions 0 and 1 willmeet at a common vertex of 1100000000. To guarantee thek paths are disjoint, each path needs a unique signature definedthrough a key-bit. A key-bit is a bit in a node address.It will be assigned to each of the k neighbors of node u i tocarry the signature that is unique to the path through thatneighbor before applying the point-to-point routing algorithmusing a hamiltonian cycle. If we say “the key-bit isat the dimension x”, it means that, negating the value of thekey-bit of a node will get the address of that nodes’ xth dimensionalneighbor. The k + m dimensions of an MC(k,m)are 0,1,...,k − 1,k,k + 1,...,k + m − 1.The key-bit can be determined as below. Notice that allthe c ui , the class_id of node u i ∈ C u , for i = 0,1,...,2 m −1,are the same, so we use c u to denote c ui . In such a case, nodeu may be any of node u i , 0 ≤ 2 m − 1. c v does also. We usec ( u j) and c ( vj) to denote c u ( j) and c v ( j), respectively. In the caseof c u = c v , if we can find a bit where M u [c ( u j) ] and M v [c ( v j) ]have the same value, then let that bit be the key-bit (type1); otherwise, take any bit as the key-bit (type 2). The ideabehind this is to enforce a signature (changing the key-bitvalue) before applying the point-to-point routing algorithm.In the case of type 1, the key-bit should be removed finally.In the case of c u ≠ c v , the construction of k disjoint pathshas two parts. The first part is the same as the case of c u = c vand the second part is to construct the subpaths that containcross-edges only, since after the first part finished, the updatingof fields M[i], 0 ≤ i ≤ 2 k −1, has been done. To guaranteethe paths are disjoint, for the second part, we shouldfind k disjoint class-paths in the k-cube. This can be doneas the same manner as a traditional hypercube does. Wesummarize it below.Let x and y be two nodes in an n-cube and the Hammingdistance between x and y, |x ⊕ y| > 1. Let Z j = (x ( j) →y), 0 ≤ j ≤ n − 1, are n disjoint paths in an n-cube, d pathsare of length (d −1) and (n−d) paths are of length (d +1),where d = |y ⊕ x| is the Hamming distance between x andy. Without loss of generality, we assume that the d paths oflength (d − 1) are (x ( j) → y (( j+1) mod d) : y), 0 ≤ j ≤ d − 1,and the (n−d) paths of length d +1 are (x ( j) → y ( j) : y), d ≤j ≤ n − 1.Let the two clusters be C u = (c u ,M u [h −1],...,M u [c u + 1],∗,M u [c u − 1],...,M u [0]) andC v = (c v ,M v [h−1],...,M v [c v +1],∗,M v [c v −1],...,M v [0]).Let HC be a hamiltonian cycle in H k . In what follows,we give an algorithm for pairing up the nodes in C u andC v and constructing k disjoint paths for each pair. In thealgorithm, the hamiltonian path that follows the directionof (u i → u ( j)i) will be used.Algorithm 2 (Multi_Channel_Cube(C u ,C v ))beginCase 1: c u = c v . We pair up nodes u i ∈ C u and v i ∈ C v sothat M ui [c u ] = M vi [c v ], for i = 0,1,...,2 m − 1. Finda 0 in M ui [c ( u j) ] ⊕ M vi [c ( vj) ] from rightmost bit. If thebit position be x, let the key-bit is at the dimensiony = k + (x mod k). Then (u i → v i ) j = (u i : u ( j)u ( j,y)i: u ( j,y,l)i→ v ( j)i: v i ), where (u ( j,y)ione-step path in HC and (u ( j,y,l)i→ v ( j)ii:: u ( j,y,l)i) is an) is a path constructedby Algorithm 1, for i = 0,1,...,2 m − 1, andj = 0,1,...,k − 1.Case 2: c u ≠ c v . We pair up nodes u i and v i so thatM ui [c u ] ⊕ M vi [c v ] = M ui [c v ] ⊕ M vi [c u ]. Let w i, j be ofclass c ( uj) and its cluster_id + node_id equal to thatof v i , i.e. w i, j differs with v i at field c only. In whatfollows, if u ( j)i= w i, j , the path u ( j)i→ w i, j will be replacedwith u ( j)i.Case 2.1: |c u ⊕ c v | = 1. Let the bit position wherec u and c v have different value be q. We havew i,q = v i in this case. The path in dimension qis (u i : u (q)iv (q)i→ v (q)i: w i,q = v i ), where (u (q)i→) is constructed by Algorithm 1. The restk − 1 paths are (u i : u ( j)i: u ( j,y)i: u ( j,y,l)i→ w i, j :: u ( j,y)iis a) is an one-step path inv ( j)i: v i ), 1 ≤ j ≤ k − 1, where u ( j)i: u ( j,y,l)isignature, (u ( j,y)iHC, and path (u ( j,y,l)(u ( j,y)i(( j+1) mod d)i→ w i, j ) is constructed byAlgorithm 1, for 0 ≤ j ≤ k − 1, j ≠ q.Case 2.2: |c u ⊕ c v | > 1. The first d paths are (u i :u ( j)i: u ( j,y)(( j+1) mod d)i→ w i, j → vi: v i ), 0 ≤ j ≤d − 1, where (u ( j)i: u ( j,y)i) is a signature, path→ w i, j ) is the path constructed by Algorithm1, and path (w i, j → vi: v i ) isconstructed by the cross-edges specified by Z j .The rest k−d paths are (u i : u ( j)i: u ( j,y)i→ w i, j →v ( j)i: v i ), d ≤ j ≤ k − 1, where (u ( j)ia signature, path (u ( j,y)i: u ( j,y)i) is→ w i, j ) is the path constructedby Algorithm 1, and path (w i, j → v ( j)i:347
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