### Table III: Energy of cube, exploiting symmetries

### Table 2: Time gain by exploiting symmetries in DSP bench- marks.

"... In PAGE 5: ... PC machine, with a 1.6GHz Athlon processor and 1.5GB RAM. Table2 shows the gain in terms of time by exploiting symme- tries in different DSP programs. The first column gives the name of the benchmark, the second indicates the basic block that is being optimized, and the third gives the size of the basic block in term of number of DAG nodes.... ..."

### Table 5.1: Filtering with the std-2x algorithm exploiting kernel symmetry.

### Table 12: Factorization time of large test matrices on the CRAY T3E. \| quot; indicates not enough memory. ( )SuperLU computes an LU factorization of matrix ship 003 and does not exploit the symmetry in the matrix.

"... In PAGE 35: ...1.1 Study of the factorization phase We show in Table12 the time spent during the factorization phase of both solvers. On a relatively small number of processors (less than 32) MUMPS is generally faster than SuperLU for two reasons.... In PAGE 37: ... This again gives an indication of the better scalability of SuperLU compared to MUMPS. To better understand the performance di erences observed in Table12 and to identify the main characteristics of our solvers, we show, in Figures 12, 13, and 14, the average communication volume on 4 and 64 processors respectively. The speed of communication can depend very much on the size of the messages and we thus indicate, in Figure 15, the average message size on 64 processors.... ..."

### Table 1. Influence of the symmetry exploitation on the size of the search tree, on the number of solutions, and on the resolution time

in Using Symmetry of Global Constraints to Speed up the Resolution of Constraint Satisfaction Problems

1998

"... In PAGE 2: ... In this case, symmetry can be exploited to compute only a subset of the whole solution set. Table1 below illustrates the influence of a symmetry detection on these two problems: Table 1. Influence of the symmetry exploitation on the size of the search tree, on the number of solutions, and on the resolution time... ..."

Cited by 8

### Table 5.3: Filtering with the dot-2x algorithm, exploiting the per-pixel dot product, computing four passes within a single pass, as well as exploiting the kernel symmetry (optional).

### Table 5: Execution times for the applications. Each system has been solved twice, once with (symmet.) and once without (no sym.) the exploitation of the symmetry. In the rst case, F 0 is a system with random coe cients to be used as start system. The timings are given for the construction of the subdivision (Subdiv.), the solution of F 0 and F . The last column contains the sum of these timings.

1995

"... In PAGE 24: ... Table 4 summerizes all characteristical gures for the appli- cations. The execution times listed in Table5 only have a relative meaning, only meant to compare the advantages of exploiting the symmetry. Acknowledgments.... ..."

Cited by 18

### Table 1: Context formation for sign encoding/decoding. We only use the sign from the two nearest neighbors for context formation. We exploit symmetry by using a sign ipping technique and we thus reduce the number of classes from nine to ve. g0; g1 are the signs of the wavelet coe cients at the corresponding locations.

"... In PAGE 16: ... ipping [19] we can reduce the number of classes from nine to ve. In sign ipping we take advantage of certain symmetries present in the context formation, as seen in Table1 . Let us consider an example.... ..."

### Table 4: The boundary conditions associated with the symmetry cell in Figure 7b. 5. AN IMPLEMENTATION OF AUTOMATED SYMMETRY EXPLOITATION The methodology discussed in the previous sections has been translated into algorithms (written in MATLAB) for 2-D solids and integrated with FEMLAB, a multi-

"... In PAGE 7: ...summarized in Table4 . Figure 10 illustrates the first eigen-mode for each of these problems.... ..."