### Table I Five graph homomorphisms and their properties

### TABLE I HOMOMORPHIC HASHING FUNCTION PARAMETERS

### Table 1: Recursion for Computing the Partition Function.

1999

Cited by 2

### Table 2 Computational complexity in homomorphic encryption Volume One bidder

"... In PAGE 3: ... Therefore, the computation and communication complexity of e-auction based on homomorphic encryption are increased in opening stage. Table2 shows the computational complexity of [CLK03] scheme based on homomorphic encryption. Table 2 Computational complexity in homomorphic encryption Volume One bidder ... ..."

### Table 4: Performance of various algorithms for performing the initial partition of the coarse graph.

1998

"... In PAGE 13: ... We have implemented the following algorithms: (a) spectral bisection (SBP), (b) graph growing (GGP), and (c) greedy graph growing (GGGP). The result of the partitioning algorithms for some matrices is shown in Table4 . These partitions were produced by using the heavy-edge matching (HEM) during coarsening and the BKL(*,1) refinement policy during uncoarsening.... In PAGE 13: ...d) the combined time (IRTime) spent in partitioning (ITime) and refinement (RTime) for the 32-way partition (i.e., IRTime = ITime + RTime). A number of interesting observations can be made from Table4 . The edge-cut of the initial partition (IEC) for the GGGP scheme is consistently smaller than the other two schemes (4ELT is the only exception as SBP does slightly better).... In PAGE 13: ... Hence, much of the difference in the run time of the three different initial partition schemes is due to refinement time associated with each. Furthermore, SBP produces partitions that are significantly worse than those produced by GGP and GGGP (as it is shown in the IEC column of Table4 ). This happens because either the iterative algorithm used to compute the eigenvector does not converge within the allowable number of iterations4, or the initial partition found by the spectral algorithm is far from a local minimum.... In PAGE 14: ... Hence, we omitted these results here. In summary, the results in Table4 show that GGGP consistently finds smaller edge-cuts than the other schemes, and even requires slightly smaller run time. Furthermore, there is no advantage in choosing spectral bisection for partitioning the coarse graph.... ..."

Cited by 495

### Table 4: Performance of various algorithms for performing the initial partition of the coarse graph.

1998

"... In PAGE 13: ... We have implemented the following algorithms: (a) spectral bisection (SBP), (b) graph growing (GGP), and (c) greedy graph growing (GGGP). The result of the partitioning algorithms for some matrices is shown in Table4 . These partitions were produced by using the heavy-edge matching (HEM) during coarsening and the BKL(*,1) refinement policy during uncoarsening.... In PAGE 13: ...d) the combined time (IRTime) spent in partitioning (ITime) and refinement (RTime) for the 32-way partition (i.e., IRTime = ITime + RTime). A number of interesting observations can be made from Table4 . The edge-cut of the initial partition (IEC) for the GGGP scheme is consistently smaller than the other two schemes (4ELT is the only exception as SBP does slightly better).... In PAGE 13: ... Hence, much of the difference in the run time of the three different initial partition schemes is due to refinement time associated with each. Furthermore, SBP produces partitions that are significantly worse than those produced by GGP and GGGP (as it is shown in the IEC column of Table4 ). This happens because either the iterative algorithm used to compute the eigenvector does not converge within the allowable number of iterations4, or the initial partition found by the spectral algorithm is far from a local minimum.... In PAGE 14: ... Hence, we omitted these results here. In summary, the results in Table4 show that GGGP consistently finds smaller edge-cuts than the other schemes, and even requires slightly smaller run time. Furthermore, there is no advantage in choosing spectral bisection for partition- ing the coarse graph.... ..."

Cited by 495

### Table 4: Performance of various algorithms for performing the initial partition of the coarse graph.

1998

"... In PAGE 13: ... We have implemented the following algorithms: (a) spectral bisection (SBP), (b) graph growing (GGP), and (c) greedy graph growing (GGGP). The result of the partitioning algorithms for some matrices is shown in Table4 . These partitions were produced by using the heavy-edge matching (HEM) during coarsening and the BKL(*,1) refinement policy during uncoarsening.... In PAGE 13: ...d) the combined time (IRTime) spent in partitioning (ITime) and refinement (RTime) for the 32-way partition (i.e., IRTime = ITime + RTime). A number of interesting observations can be made from Table4 . The edge-cut of the initial partition (IEC) for the GGGP scheme is consistently smaller than the other two schemes (4ELT is the only exception as SBP does slightly better).... In PAGE 13: ... Hence, much of the difference in the run time of the three different initial partition schemes is due to refinement time associated with each. Furthermore, SBP produces partitions that are significantly worse than those produced by GGP and GGGP (as it is shown in the IEC column of Table4 ). This happens because either the iterative algorithm used to compute the eigenvector does not converge within the allowable number of iterations4, or the initial partition found by the spectral algorithm is far from a local minimum.... In PAGE 14: ... Hence, we omitted these results here. In summary, the results in Table4 show that GGGP consistently finds smaller edge-cuts than the other schemes, and even requires slightly smaller run time. Furthermore, there is no advantage in choosing spectral bisection for partitioning the coarse graph.... ..."

Cited by 495

### Table 1.1 Summary of NP- and #P-hardness results for planar instances. The third column summarizes the decision complexity of the problems while the fourth column summarizes the complexity of the counting versions. A star (*) denotes a result obtained in this paper. The numbers in square brackets are the references where the corresponding results are proved.

### Table 1: Various graphs used in evaluating the parallel multilevel k-way graph partitioning al- gorithm.

1998

"... In PAGE 11: ... However, due to the k-way refinement performed in the uncoarsening phase, the final partitions are only slightly worse than those produced by the serial k-way algorithm (that uses the multilevel recursive bisection al- gorithm for computing initial partitions). Partition Quality Table 2 shows the quality of the partitions produced by the parallel k-way al- gorithm as well as the amount of time it took to produce these partitions on a Cray T3D for the problems of Table1 . Partitions for in 16, 32, 64, and 128 parts are shown, each produced on 16, 32, 64, and 128 processors, respectively.... ..."

Cited by 333

### Table 3: Relations between the numbers of one-pass improvement iteration (outside repeat of the graph partitioning algorithm) and the graph partitioning.

in A graph-based clustering method for a large set of sequences using a graph partitioning algorithm

2001

"... In PAGE 8: ...nd 7.9Mbytes memory. Computation for the second or the other components take very few time and memory. Table3 shows the number of one-pass improvement iteration (outside repeat of our graph partitioning algorithm) and the number of graph partitioning for the largest connected components and the sum of all components. It reveals that the number of one-pass improvement iteration is less than five in most cases.... ..."

Cited by 4