### Table 5: Perturbation cost in relation to the arithmetic operations

### Table 2 Results of the multistart algorithm using the unidirectional perturbation-based cost N=2 N=5

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### Table 4 Results of the crude and ARS algorithms using the unidirectional perturbation- based cost

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### Table 10 Results of the SCE algorithm using the unidirectional perturbation-based cost 5 Complexes 10 Complexes

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### Table 12 Results of the multistart algorithm using the unidirectional perturbation-based cost N=3 N=5

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### Table 3: Cost equivalence classes for a subset of 255 mutation functions. Perturbation index Mutation index Mutation function Mutation Cost

"... In PAGE 9: ... Classes with ($i = 4; $i = 2; $i = ?1) share a total of 140 mutation functions and have an important role: they always bind both wires within Pj to one or two nodes (creating reconvergence for lower level nodes), 60 out of 140 mutations reduce the cutset size, 40 out of 140 mutations also prune logic nodes in the forward path. A subset of 255 mutation functions is shown in Table3 . The full set is available in [27].... ..."

### Table 8 Results of the CRS algorithm using the unidirectional perturbation-based cost 2(N + 1) 10(N + 1)

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### Table 1. The number of sweeps, SU, cost per vertex, v, and total cost, t, to untangle the perturbed meshes H = 1 H = 2 H = 4 H = 8 Method

"... In PAGE 6: ... The resulting initial meshes contain 256, 288, 301, and 312 inverted elements, respectively. In Table1 , we give results for each of the algo- rithms described in Section 3. In particular, the... In PAGE 8: ... The cost per vertex in each of these methods falls between the IT-LS and 3T methods, with the heavier use of the 3T method evident in the higher cost of the CA1 al- gorithm. In the last two rows of Table1 , we show the results of preceding CA2 with three and ten sweeps of Laplacian smoothing, respectively. In all cases, Laplacian smoothing signi cantly reduces the number of optimization iterations, and conse- quently both the cost per vertex and total cost, re- quired to untangle the mesh.... In PAGE 8: ... This mesh clearly has the best quality, and Laplacian smoothing, although it can not be used alone, is highly recommended as a preconditioner for the optimization-based un- tangling methods. In Figure 7, we show the convergence history of each of the methods listed in Table1 . In partic- ular, we give the number of invalid quadrilateral elements that remain after each sweep through the mesh for the H = 4 case.... ..."

### Table 1. Variation of the running time and cost of Industry-1 with various perturbation coe cient running time speedup over speedup over change % from change % from

1996

"... In PAGE 2: ...f 180 and a quality degradation of only 10.0 %. As stated in the analysis section, after perturbation, the regularity of the co- e cient matrix is totally changed. For the circuit Industry-1, compared with PROUD, the speedup of the modi ed PROUD is quite signi cant as shown in Table1 . For example when = 0:05 the overall speedup is 5:08: But the cost increase is only about 3.... ..."

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### Table 2: Perturbed Escape Time Problem Errors

"... In PAGE 19: ... Now, we de ne c(x) on the regions of strong regularity by c(x) = (1=2)DDV 0(x); a(x)DV 0(x)E ? Db(x); DV 0(x)E: Our use of a molli er in de ning all of the relevant functions ensures that the cost function c(x) extends smoothly to c(x) = 1=2 at the singularities, and it turns out that V 0(x) solves the limit control problem for the indicated cost structure. In Table2 , we indicate the errors in computing the value function and the optimal controls using the three numerical schemes considered in this paper. Errors are given for the entire domain and for those regular points which are at least a distance of 0:1 away from the singularities.... ..."