### Table 1: Optimal node-disjoint paths in SG,

### Table 1: Optimal node-disjoint paths in SGn SGn has a vertex connectivity of k(SGn) = (n?1), and therefore can tolerate up to (n ? 2) node failures. Exact values for the fault-diameter of the star graph were derived in [7] and are given below. This result is obtained via a worst-case analysis that considers the impact of up to (n?1) arbitrary faults on the optimal- length node-disjoint paths listed in Table 1. df(SGn) = d(SGn) + 1; if n is odd or n 7 d(SGn) + 2; if n = 4 or n = 6

"... In PAGE 4: ... The rst step in the method consists of selecting a permutation requiring a large number of lateral and local links in the path from u to v. The number of lat- eral links depends on the number of cycles of length at least 2 in (c) and on the number of digits in these cycles (m), as shown in Table1 . The number of lo- cal links also depends on c and m, but can be further increased by a proper selection of the internal compo- sition of the cycles in .... In PAGE 5: ... Initially, we consider some candidates for permutation that are likely to result in the largest possible number of links in the presence of faults. The values of c and m in the selected permutations result in the highest or close to the highest possible number of lateral links along the di erent routes from to the identity node (see Table1 ). Naturally, this selection criterion also increases the number of local links in these routes.... In PAGE 8: ... Due to space constraints, we show in Table 10 only the sequences of lateral links R(`1 7! `s) used to build Table 9d. One possible worst-case fault placement that can be applied to Table 9d is shown in Table1 1a. The costs of the optimal paths for all possible combinations of digits i and j are listed in Table 11b.... In PAGE 8: ... In. Final link link 2 3 4 5 6 2 { 18 19 19 18 3 { { { { { 4 { 20 19 20 19 5 { 19 20 20 19 6 { 20 20 20 19 (a) Worst-case fault placement for = (1 3 5 2)(4 6) i j 2 3 4 5 6 2 19 18 19 19 18 3 20 19 20 20 19 4 20 20 19 20 19 5 20 19 20 20 19 6 20 19 20 20 19 (b) Cost of optimal paths under the fault placement of Table 11a Table1 1: A worst-case fault placement in SCC6 Note that the cost of the longest path in Table 11b is 20. Hence, df(SCC6) = d(SCC6) + 1 = 20.... In PAGE 9: ... Final link link 2 3 4 5 6 2 19 18 19 19 18 3 14 22 16 19 15 4 15 20 19 20 19 5 18 19 20 20 19 6 16 20 20 20 19 (d) = (1 3 5 2)(4 6) Table 9: Cost of paths Q(`1 7! `s) in SCC6 Init. Final link link 2 3 4 5 6 2 (2; 6; 4; 6; 2; 3;5;2) (2; 3; 4; 6; 4; 5; 2; 3) (2; 3; 5; 2; 3;4;6; 4) (2; 6; 4; 6; 5; 2; 3;5) (2; 6; 4; 5; 2;3;5;6) 3 (3; 4; 6; 4; 5;2) (3; 4; 6; 4; 5;3; 2; 3; 2;3) (3; 5; 2; 4; 6;4) (3; 4; 6; 4; 2; 5; 2;5) (3; 5; 2; 6; 4;6) 4 (4; 6; 4; 3; 5;2) (4; 6; 4; 5; 2; 3; 5; 3) (4; 6; 5; 2; 4;3;5; 4) (4; 6; 4; 3; 2; 5; 2;5) (4; 3; 5; 2; 6;4;3;6) 5 (5; 6; 4; 5; 3; 5;6;2) (5; 6; 4; 6; 2; 3; 5; 3) (5; 2; 3; 5; 3;4;6; 4) (5; 6; 4; 6; 2; 5; 3;5) (5; 2; 6; 4; 5;3;5;6) 6 (6; 4; 6; 3; 5;2) (6; 4; 6; 5; 2; 3; 5; 3) (6; 5; 2; 4; 6;3;5; 4) (6; 4; 6; 2; 5; 2; 3;5) (6; 4; 5; 2; 6;3;5;6) Table1 0: Sequences of lateral links R(`1 7! `s) used to build Table 9d to the diameter of a fault-free SCCn?2 graph by the recurrence below. This recurrence holds for n 6 and can be veri ed from Equation 1.... ..."

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### Table 2: Mesh coordinate o sets used in the packing of Q(5) into S(6) 3.2 Packing Q(n ? 2) into S(n) Theorem 2 It is possible to pack pn?2 node-disjoint copies of Q(n ? 2) into S(n), with load 1, dilation 3, and expansion X(1; 3; pn?2), where pn?2 = 3 jn2 k! n ? 1 2

"... In PAGE 10: ...ode-disjoint submesh. This o set is added to m[ ], completing the mapping onto M(n ? 1). The resulting mesh coordinate is nally mapped onto S(n) via a call to Algorithm 1. Table2 shows the o set vectors that are required to pack Q(5) into S(6). Note that according to Theorem 1 it is possible to pack 12 copies of... ..."

### TABLE 1. A summary of lengths of node-disjoint paths in Cases 1 5. Not insert Insert

2000

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### Table 2 shows upper bounds for the number of messages originated in one round by two clock synchronization algorithms: 1) m-ICV, and 2) a unistep ICV algorithm employing the TCID com- munication protocols of [27] and any of the convergence functions in [14, 16, 22, 28, 43]. These bounds are respectively derived in [6] and [27], under the assumption that (M) node-disjoint paths are used to convey clock values reliably, where (M) is the connectivity of the network. Note that the numbers of messages in arbitrary-topology networks should be regarded as absolute upper bounds for both m-ICV and TCID ICV. In regular NCCNs such as tori and hypercubes, fewer messages are used by both m-ICV and TCID ICV. In all cases, m-ICV uses orders of magnitude fewer messages than TCID ICV. For example, the TCID ICV algorithm uses about 1; 000 times the number of messages of 3-ICV in a 32,768-node hypercube.

1998

"... In PAGE 25: ... Table2 : Number of messages per round (upper bounds) Another interesting aspect of m-ICV is that it increases signi cantly the communication locality for clock synchronization messages in regular NCCNs. As an example, assume that 2-ICV is used to synchronize an N-node hypercube, in which TSPs are mapped to a 2(b(log2 N)=2c) 2(d(log2 N)=2e) array.... In PAGE 32: ...Figure 1: Synchronization of groups via multistep interactive convergence Figure 2: Synchronization of a 16-node system with 2-ICV (C = clock, A = a-clock) Table 1: Trade-o between fault tolerance and communication cost (m = 2, N = 256) Figure 3: The jth round of m-ICV Figure 4: Algorithm m-ICV pseudocode Table2 : Number of messages per round (upper bounds)... ..."

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### Table 2: Algorithm for Subgraph Sampling

2001

"... In PAGE 8: ... Sampling entire subgraphs preserves the association between each core object and all the peripheral objects neces- sary for accurate calculation of attributes. Table2 lists a generic algorithm for subgraph sampling. The algorithm first assigns sub- graphs to prospective samples, and then incrementally converts prospective assignments to permanent assignments only if the subgraphs are separated from subgraphs already assigned to samples.... In PAGE 10: ...The algorithm for subgraph sampling ( Table2 ) depends on the predicate separate(Si,Sj) which indicates whether two subgraphs consist of disjoint sets of objects. We differenti- ate among three criteria for determining subgraph separation.... ..."

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### Table 5.1: Algorithm LPE returns a set of node pairs that constitute the endpoints of k-node paths in the input graph. The subgraphs GB and GW represent the black and white subgraphs induced by the current random coloring. Intuitively, we hope that we can detect a k-node path by nding its rst half in GB and its second half in GW recursively. Since the probability for its k nodes to be colored in this fashion is 2 k, it takes about 2k colorings to obtain a constant success probability.

### Table 3. Mesg. requirements in hypercubes

in Fault-Tolerant Clock Synchronization of Large Multicomputers via Multistep Interactive Convergence

1996

"... In PAGE 9: ... In this case, clock values can be conveyed along node-disjoint paths whose length is at most d(log2 N)=2e+2. Table3 compares 2-ICV with TCID ICV from the viewpoint of number of messages and maximum path length, for hypercubes of various sizes.... ..."

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### Table 1: Number of backup lightpaths generated by 1+1 protection and by DSP with different connectivity levels k.

2005

"... In PAGE 8: ... Thereby, the required number of backup lightpaths can be kept as low as possible. Table1 provides for a range of demand values d the number of backup lightpaths needed for 1+1 protection (i.... In PAGE 9: ... Hence DSP lies in between 1+1 and shared protection. Compared to 1+1 protection, Table1 shows that the number of required backup lightpaths can on the one hand be significantly reduced for meshed networks with higher connectivity. On the other hand, spreading the routing on node-disjoint paths for a demand yields in general longer lightpaths and thus increases the capacity needed for the normal operation routing.... In PAGE 12: ... Note that, for scenario full protected in case of 2-connectivity the total number of lightpaths to route equals the number of 1+1 protection (as pointed out in Section 3). Also note that for sce nario ^-protected the total number of lightpaths does not differ between 2-connectivity and maximum connectivity since d attains the same maximum for all d, see Table1 . By Table 1, we also can derive that 13 demands have value 1, since only those demands cause an increase of d in comparison with d for ^-protected.... In PAGE 12: ... Also note that for sce nario ^-protected the total number of lightpaths does not differ between 2-connectivity and maximum connectivity since d attains the same maximum for all d, see Table 1. By Table1 , we also can derive that 13 demands have value 1, since only those demands cause an increase of d in comparison with d for ^-protected. 4.... ..."

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### Table 5. Comparison with the error-correcting sub-graph isomorphism algorithm

2002

"... In PAGE 5: ... This can be very significant when matching large graphs. Table5 shows a comparison with the A*- based error-correcting algorithm over 1000 pairs of graphs generated randomly. The size of each graph is between 2 and 10 nodes.... ..."

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