### Table 4.5: Number of DBs d and test length t as function of L = 2k Faults Number of DBs Test length L = 2k (d/t)

2000

### 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 1. Summary of tracking errors

"... In PAGE 6: ...ame activation value to all winners (aj=1 for all j=1..k). In this case we would be improving the fault-tolerance and the total accuracy but worsening the tolerance to noise and the generalization ability of the proposed model. Table1 summarizes the tracking errors for the trajectories discussed in this work. The best condition refers to the situation in which the first winners for each state of the trajectories are used to retrieve them.... ..."

### Table 1. Summary of tracking errors

"... In PAGE 6: ...ame activation value to all winners (aj=1 for all j=1..k). In this case we would be improving the fault-tolerance and the total accuracy but worsening the tolerance to noise and the generalization ability of the proposed model. Table1 summarizes the tracking errors for the trajectories discussed in this work. The best condition refers to the situation in which the first winners for each state of the trajectories are used to retrieve them.... ..."

### Table 1: Our results and their comparison with previous work. The parallel model of computation is the arbitrary CRCW PRAM [8]. The second contribution is based on the following observation: The results in Table 1 require a priori the knowledge of the arboricity of the input graph. Since computing the exact value of arboricity seems to be hard [12, 18], we provide here algorithms that compute a 2-approximation for arboricity (i.e., an approximation which can be at most twice the exact value). Moreover, we show that using the approximate value, we can still obtain an optimal implicit representation of a sparse graph. The k-forest coloring problem is of independent interest since it is a fundamental prob- lem in the design of fault-tolerant communication networks [7], analysis of electric networks [6, 15] and the study of rigidity of structures [11]. 2 Preliminaries We rst show that an optimal implicit representation of a graph G can be obtained opti- mally, if a k-forest coloring of G is given.

"... In PAGE 3: ... First, we provide optimal sequential and parallel algorithms for obtaining optimal implicit representations of sparse graphs. Our results and their comparison with previous work are summarized in Table1 . It is worth noting that our results are achieved by simple and rather intuitive techniques compared with those used in [3, 4, 17] and moreover, our algorithms are easy to implement.... In PAGE 7: ... Hence, the total resource bounds are as those stated in the theorem. 4 Approximating Arboricity The results listed in Table1 require a priori the knowledge of the arboricity of the input graph in order to obtain its optimal implicit representation. However, the known algorithms for computing the exact value of the arboricity are based on matroid theory: a sequential algorithm [5] and a randomized parallel algorithm [12].... ..."

### Table 2: Degree of fault tolerance and energy consumption for B4.

"... In PAGE 6: ... We consider 4 schemes in our simulation: (1) without checkpointing and DVS (S1), (2) with checkpointing but without DVS (S2), (3) without checkpointing and with DVS (S3), and (4) with checkpointing and DVS (S4). The degree of fault tolerance and energy consumption for the office-automation benchmark (B4) is shown in Table2 . We consider the AMD K6 processor in our simulation.... In PAGE 6: ... First, we note that the proposed checkpointing schemes improve the degree of fault tolerance. As seen from Table2 , S2 and S4 can tolerate more faults than S1 and S3. Second, as expected, DVS saves energy in a distributed real-time embedded system.... ..."

### Table 7. Control statistics for optimised variable topology controllers with component and plan fault-tolerance Fitness

"... In PAGE 7: ...8 Time (s) St r o k e ( mm) Figure 15. Step response for best evolved fault-tolerant controller (thick black curve shows the response without failure, and other curves show the response with each fault) Five GA runs were done using each of the fitness functions, and the fittest results are presented in Table7 and Figures 14-15. The results show that the controllers robust to plant failure do not perform as well as the other evolved controllers.... ..."

### Table 1: Performance of PIR and FT-PIR (all in milliseconds)

2002

"... In PAGE 13: ...91%) of TPT. As shown in Table1 , the performance overhead introduced by FT-PIR is about 6% when f = 1 and K = 3, in comparison with the corresponding PIR implementation. In order to tolerate up to two faults, the FT-PIR imposed about 26% overhead.... ..."

Cited by 3

### Table 1 E ect of Problem Size on Convergence Rate Note: Each two-grid MGCG method is implemented twice for a sample problem: once on a small mesh of 17;289 unknowns and once on a larger mesh of 126;225 unknowns. Variants of MGCG di er from each other in the number of coarse-grid relaxation passes. nl=ns is a measure of independence of each method from mesh size. Iterative

"... In PAGE 16: ...2, we were able to run the two-grid MGCG methods on two di erent mesh discretizations of the same problem. Table1 shows the number of iterations, ns and nl, necessary for the various methods (including DCG and ICCG) to reach convergence on the small problem (17,289 unknowns) and on the \large quot; problem (126,225 unknowns), respectively. For a method truly independent of mesh size, the ratio of these two, nl=ns, would be 1.... ..."

### Table 5. Number of messages for a k-AMT (M=4) with 64, 256, 1024 leaves and k=1 (k= 0).

1997

"... In PAGE 21: ...n order to tolerate as many faults as those calculated in the previous Section 3.3.1, does not exceed the 81% of the latency experienced by a frame over the same k-AMT architectures without redundancy. Finally, Table5 shows the total number of messages transmitted by the k-AMTS over the three different k-AMT architectures introduced above. The total number of messages for the same k-AMT architectures without link redundancy (i.... ..."

Cited by 10