### Table 1. Performance Guarantees for finding spanning trees in an arbitrary graph on n nodes. Asterisks indicate results obtained in this paper. gt; 0 is a fixed accuracy parameter.

"... In PAGE 3: ... There he provides an approximation algorithm for the (Degree, Diameter, Spanning tree) problem with performance guarantee (O(log2 n); O(log n))6. The (Diameter, Total cost, Spanning tree) entry in Table1 corresponds to the diameter-constrained minimum spanning tree problem introduced earlier. It is known that this problem is NP-hard even in the special case where the two cost functions are identical [HL+89].... In PAGE 5: ... 3.1 General Graphs Table1 contains the performance guarantees of our approximation algorithms for finding spanning trees, S, under different pairs of minimization objectives, A and B. For each problem cataloged in the table, two different costs are specified on the edges of the undirected graph: the first objective is computed using the first cost function and the second objective, using the second cost function.... In PAGE 5: ... For example the entry in row A, column B, denotes the performance guarantee for the problem of minimizing objective B with a budget on the objective A. All the results in Table1 extend to finding Steiner trees with at most a constant factor worsening in the performance ratios. For the diagonal entries in the table the extension to Steiner trees follows from Theorem 6.... ..."

### Table 1. Performance Guarantees for finding spanning trees in an arbitrary graph on n nodes. Asterisks indicate results obtained in this paper. gt; 0 is a fixed accuracy parameter.

"... In PAGE 3: ... There he provides an approximation algorithm for the (Degree, Diameter, Spanning tree) problem with performance guarantee (O(log2 n); O(log n))6. The (Diameter, Total cost, Spanning tree) entry in Table1 corresponds to the diameter-constrained minimum spanning tree problem introduced earlier. It is known that this problem is NP-hard even in the special case where the two cost functions are identical [HL+89].... In PAGE 5: ... 3.1 General Graphs Table1 contains the performance guarantees of our approximation algorithms for finding spanning trees, S, under different pairs of minimization objectives, A and B. For each problem cataloged in the table, two different costs are specified on the edges of the undirected graph: the first objective is computed using the first cost function and the second objective, using the second cost function.... In PAGE 5: ... For example the entry in row A, column B, denotes the performance guarantee for the problem of minimizing objective B with a budget on the objective A. All the results in Table1 extend to finding Steiner trees with at most a constant factor worsening in the performance ratios. For the diagonal entries in the table the extension to Steiner trees follows from Theorem 6.... ..."

### Table 6. Sparse graphs

1996

"... In PAGE 17: ... We increased the number of nodes and de ned the number of edges to be 1.5 times the number of nodes and 2 times jV j (see Table6 ) for the unweighted case. Computational studies on randomly generated unweighted graphs showed that our algorithm can solve almost all instances of graphs with at most 40 edges.... ..."

Cited by 34

### Table 6. Sparse graphs

1996

"... In PAGE 17: ... We increased the number of nodes and de ned the number of edges to be 1.5 times the number of nodes and 2 times jV j (see Table6 ) for the unweighted case. Computational studies on randomly generated unweighted graphs showed that our algorithm can solve almost all instances of graphs with at most 40 edges.... ..."

Cited by 34

### Table 5 Solution length using max-degree abstraction. Radius of Abstraction

1996

"... In PAGE 43: ... The exact radius at which this degeneration occurs varies from graph to graph, and depends primarily on the graph apos;s diameter (maximum distance between two nodes). All but two of the graphs have small diameters (an approximate indication of the diameter is the average optimal solution length in Table5 ), and abstraction of the small diameter graphs leads to little or no speedup for the larger radii.A radius of 2 maximizes the speedup over breadth first search of every refinement technique on every graph (except Towers of Hanoi, where a radius of 3 maximizes speedup).... ..."

Cited by 31

### Table 1: Various graphs used in evaluating the multilevel graph partitioning and sparse matrix ordering algorithm.

1998

"... In PAGE 11: ... 3 Experimental Results We evaluated the performance of the multilevel graph partitioning algorithm on a wide range of graphs arising in different application domains. The characteristics of these graphs are described in Table1 . These graphs are classified into six groups.... ..."

Cited by 333

### Table 1: Various graphs used in evaluating the multilevel graph partitioning and sparse matrix ordering algorithm.

1998

"... In PAGE 11: ... 3 Experimental Results We evaluated the performance of the multilevel graph partitioning algorithm on a wide range of graphs arising in different application domains. The characteristics of these graphs are described in Table1 . These graphs are classified into six groups.... ..."

Cited by 333

### Table 1 : Tensile shear loads and nugget diameters for various conditions

"... In PAGE 11: ... The average and standard deviation values of the test results for these trials are considered for evaluation. Table1 summarizes the tensile shear load and nugget diameter results.... In PAGE 15: ...Tensile shear test and nugget diameter measurement results: Table1 shows the tensile shear test and nugget diameter measurement results. It is found that with poor fit up of 1mm, the tensile shear load marginally increases from 606 kgf to 622 kgf with feedback control ON condition.... In PAGE 18: ...Tensile shear test and nugget diameter measurement results: Table1 shows the tensile shear test results and nugget diameter results. It is found that with high secondary impedance, the tensile shear load marginally decreases from 606 kgf to 605 kgf with feedback control ON condition.... In PAGE 20: ... 12. : Influence of surface condition on dynamic resistance (1mm+1mm low carbon steel) - Fuzzy controller Tensile shear test and nugget diameter measurement results: Table1 shows the tensile shear test results and nugget diameter results. It is found that with rusted surfaces, the tensile shear load marginally increases from 606 kgf to 630 kgf with control on condition.... In PAGE 26: ...(B) Model Parameters : The magnitudes of L, C, R and C1 that enter into the equivalent electrical network of the vibrating quartz crystal depend upon the way in which the crystal is cut, its size and the type of vibrations involved. Numerical values can be calculated when these factors are known and have of which those listed in Table1 . are for a typical quartz crystal [8].... In PAGE 26: ...arallel resonance........430.1 kc Table1 : Characteristics of a Typical Quartz Crystal 3. EXPERIMENTAL SETUP Electrical circuits involving piezoelectric crystals can be therefore analyzed by replacing the crystal with its model i.... In PAGE 54: ... Following conclusions can be drawn from these figures: a) with increase in fiber radius increases, sensitivity increases and maximum output decreases; b) with increase in pitch increases, output decreases; c) for same value of pitch and former radius, output is higher for lesser number of turns and sensitivity is more for higher number of turns. Table1 -3 shows the variation of output power with weight for fixed values of former radius and number of turns, varying from 5 to 7. The analysis of results shows that output is an.... In PAGE 56: ...1 13.3 Table1 : Variation of output power with weight for number of turns of fiber N= 5,6 and 7 and radius of curvature Rc=7.... ..."

### Table 2: Sparse graphs: running time of various algorithms in seconds

### Table 1 Various matrices used in evaluating the multilevel graph partitioning and sparse matrix ordering algorithm.

1998

"... In PAGE 13: .... Experimental results|Graph partitioning. We evaluated the perfor- mance of the multilevel graph partitioning algorithm on a wide range of graphs arising in di erent application domains. The characteristics of these matrices are described in Table1 . All the experiments were performed on an SGI Challenge with 1.... In PAGE 18: ... The re nement policies that we evaluate are (a) KL(1), (b) KL, (c) BKL(1), (d) BKL, and (e) the combination of BKL and BKL(1) (BKL(*,1)). The result of these re nement policies for computing a 32-way partition of graphs corresponding to some of the matrices in Table1 is shown in Table 5. These partitions were produced by using the HEM during coarsening and the GGGP algorithm for initially partitioning the coarser graph.... In PAGE 20: ... Note that MSB is a signi cantly di erent scheme than the multilevel scheme that uses spectral bisection to partition the graph at the coarsest level. We used the MSB algorithm in the Chaco [25] graph partitioning package to produce partitions for some of the matrices in Table1 and compared the results with the partitions produced by our multilevel algorithm that uses HEM during coarsening phase, GGGP during partitioning phase, and BKL(*,1) during the uncoarsening phase. Figure 3 shows the relative performance of our multilevel algorithm compared with MSB.... In PAGE 26: ... Therefore, when the factorization is performed in parallel, the better utilization of the processors can cause the ratio of the runtime of parallel factorization algorithms ordered using MMD and that using MLND to be substantially higher than the ratio of their respective operation counts. The MMD algorithm is usually two to three times faster than MLND for ordering the matrices in Table1 . However, e orts to parallelize the MMD algorithm have had no success [14].... ..."

Cited by 495