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8.3. Second Experiments on Small Synthetic Graphs
87 Communication volume is a different story. Figure 8.11 shows that the 2D and 3D square graphs (sm2d100p and sm3d100p) and the 3D triangular square graph (tsm3d100p) have a clear benefit from DSHEM; it is stable for all values of -dshem_p2.
Communication volume
DSHEM (Evaluating) versus SHEM (Reference)
Figure 8.11. DSHEM vs. SHEM: effect of refinement on the communication volume with synthetic graphs. Partitioning objective: edge cut on the left, communication volume on the right.
Maximum communication volume of all subdomains DSHEM (Evaluating) versus SHEM (Reference)
Figure 8.12. DSHEM vs. SHEM: effect of refinement on the maximum communication volume of all subdomains with synthetic graphs. Partitioning objective: edge cut on the left, communication volume on the right.
Regarding the maximum communication volume of all subdomains, Figure 8.12 shows that the 2D square graph (sm2d100p) keeps an almost constant positive behavior. The 3D counterpart performs better with -dshem_p2 values lower than 100.
Execution Time
The graphs for these experiments are small enough to make virtually impossible to evaluate the impact of DSHEM on the execution time. The majority of the running times does not even reach one second.
8.3. Second Experiments on Small Synthetic Graphs
This particular set of experiments uses the graphs presented in Table 7.7 of Chapter 7. It is a set of small
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synthetic graphs which are designed to evaluate the performance of DSHEM and compare it with SHEM and Random.
8.3.1. Execution Parameters
Two main parameters are used to tune up DSHEM for this set of experiments, namely maxvwtm and -dshem_p2. The other two parameters have a fix value of 100. The values chosen for the second set of experiments are presented in Table 8.2. This particular set produces 441 different combinations of values, giving a more focused view of the performance of DSHEM. This set of experiments is designed based on the results of the first set.
Table 8.2: DSHEM parameters for the second set of small synthetic graphs.
-maxvwtm -dshem_p1 -dshem_p2 -dshem_p3
140 to 160 100 90 to 110 100
8.3.2. Analysis of Results
The experimental results presented in this section are organized in a manner to understand how the different execution parameters affect the partitions. Based on the results of the first set of experiments, the effect of the multiplier maxvwtm is evaluated with greater detail, as well as the percentage -dshem_p2. Percentages -dshem_p1 and -dshem_p3 are set to 100 as they do not influence the outcome.
Again, the robustness of DSHEM is also evaluated with different degrees of irregularity introduced to the synthetic graphs. The refinement and its influence on DSHEM are also studied. Finally, the execution time is also examined to estimate the degradation, if any, brought by DSHEM.
The analysis is carried out with the two partitioning objectives available in METIS: cut and vol; the edge cut and the total communication volume respectively. Only three metrics are presented in this thesis: total edge cut, total communication volume, and maximum communication volume of all subdomains.
Multiplier -maxvwtm
A closer analysis of the multiplier -maxvwtm brings a different scenario of that from the first set of experiments. Reducing its value produces more balanced initial partitions and the refinement process is also optimized. However, a balanced partition does not necessarily mean a smaller edge cut or reduction in communication volume; it only means that the subdomains are more equal in size.
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Edge cut
DSHEM (Evaluating) versus SHEM (Reference)
Figure 8.13. DSHEM vs. SHEM: effect of -maxvwtm on the edge cut with synthetic graphs and communication volume as partitioning objective.
Communication volume
DSHEM (Evaluating) versus SHEM (Reference)
Figure 8.14. DSHEM vs. SHEM: effect of -maxvwtm on the communication volume with synthetic graphs and communication volume as partitioning objective.
Maximum communication volume of all subdomains DSHEM (Evaluating) versus SHEM (Reference)
Figure 8.15. DSHEM vs. SHEM: effect of -maxvwtm on the maximum communication volume of all subdomains with synthetic graphs and communication volume as partitioning objective.
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From the results in Figure 8.13, Figure 8.14 and Figure 8.15, it is not possible to find a discernible pattern as suggested from the first set of experiments. It is evident that the multiplier -maxvwtm plays a role, but only after trial and error it would be possible to adapt it to specific needs.
It is also confirmed that the type of graph has an influence, being the 3D square graph (sm3d100p) and the 3D triangular square graph (tsm3d100p) with the highest improvements, and the 2D triangular square graph (tsm2d100p) and the 3D dense triangular square graph (dtsm3d100p) with the worst results.
Percentages -dshem_p1, -dshem_p2 and -dshem_p3
Percentages -dshem_p1 and -dshem_p3 are excluded from a deeper analysis as previous results suggest they do not play a role at all in the partitioning process. Percentage -dshem_p2 is used to modify the behavior of the cost function in DSHEM and improve the partition.
Edge cut
DSHEM (Evaluating) versus SHEM (Reference)
Figure 8.16. DSHEM vs. SHEM: effect of -dshem_p2 on the edge cut with synthetic graphs and communication volume as partitioning objective.
Communication volume
DSHEM (Evaluating) versus SHEM (Reference)
Figure 8.17. DSHEM vs. SHEM: effect of -dshem_p2 on the communication volume with synthetic graphs and communication volume as partitioning objective.
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Maximum communication volume of all subdomains DSHEM (Evaluating) versus SHEM (Reference)
Figure 8.18. DSHEM vs. SHEM: effect of -dshem_p2 on the maximum communication volume of all subdomains with synthetic graphs and communication volume as partitioning objective.
Depending on the metric and partitioning objective, percentage -dshem_p2 has different effects on the output of the partitioning process. Figure 8.16 shows that the influence is more erratic for the edge cut;
being the 3D square graph (sm3d100p) and the 3D triangular square graph (tsm3d100p) the most affected. With communication volume in mind, the same types of graphs benefit from the percentage -dshem_p2 being lower than 100; see Figure 8.17. Finally, when the maximum communication volume of all subdomains is evaluated, Figure 8.18 shows that values over 100 benefit some graph types.
Graph Irregularity
The performance of DSHEM is also studied with irregular graphs. Sizes 𝑛 to 𝑑, in Figure 8.19, Figure 8.20 and Figure 8.21, represent the four biggest 2D and all 3D graphs of the set.
Edge cut
DSHEM (Evaluating) versus SHEM (Reference)
Figure 8.19. DSHEM vs. SHEM: effect of irregularity on the edge cut with synthetic graphs and communication volume as partitioning objective.
From the results of the second set of experiments, it is possible to conclude that the results are consistent with the first set or experiments. One of the most stable types is the 2D dense triangular square graph (dtsm2d) when evaluating the edge cut. The introduced regularity in the graphs clearly has an impact, but the amount of irregularity does not reflect a proportional impact on the quality of the
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Communication volume
DSHEM (Evaluating) versus SHEM (Reference)
Figure 8.20. DSHEM vs. SHEM: effect of irregularity on the communication volume with synthetic graphs and communication volume as partitioning objective.
Maximum communication volume of all subdomains DSHEM (Evaluating) versus SHEM (Reference)
Figure 8.21. DSHEM vs. SHEM: effect of irregularity on the maximum communication volume of all subdomains with synthetic graphs and communication volume as partitioning objective.
With respect to the communication volume and maximum communication volume of all subdomains, similar behavior can be seen. The irregularity impacts the final partition, but it does not degrade or improves it with a clear pattern.
Refinement
To analyze the effect of the refinement process on the partitions produced by DSHEM, METIS is executed without it; without the refinement process no objective is optimized.
Figure 8.22 shows similar results as those from the first set of experiments: the 2D and 3D square graphs (sm2d100p and sm3d100p) benefit from -dshem_p2 with values from 100 and higher. However, the 3D triangular square graph (tsm3d100p) presents a less visible pattern as that from the first set. The 3D dense triangular square graph (dtsm3d100p) is also affected.
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Edge cut
DSHEM (Evaluating) versus SHEM (Reference)
Figure 8.22. DSHEM vs. SHEM: effect of refinement on the edge cut with synthetic graphs and communication volume as partitioning objective.
Communication volume
DSHEM (Evaluating) versus SHEM (Reference)
Figure 8.23. DSHEM vs. SHEM: effect of refinement on the communication volume with synthetic graphs and communication volume as partitioning objective.
Maximum communication volume of all subdomains DSHEM (Evaluating) versus SHEM (Reference)
Figure 8.24. DSHEM vs. SHEM: effect of refinement on the maximum communication volume of all subdomains with synthetic graphs and communication volume as partitioning objective.
Communication volume remains quite stable. Similar as the first set of experiments, Figure 8.23
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shows that the 2D and 3D square graphs (sm2d100p and sm3d100p) and the 3D triangular square graph (tsm3d100p) benefit from DSHEM. The percentage -dshem_p2 has an influence too, with an inflection point in 100.
With respect to the maximum communication volume of all subdomains, Figure 8.24 shows that the 2D square graph (sm2d100p) still keeps an almost constant positive benefit. The 2D triangular square graph (tsm2d100p) performs better with -dshem_p2 values lower than 100.
Execution Time
The graphs for these experiments are small enough to make virtually impossible to evaluate the impact of DSHEM on the execution time. The majority of the running times does not even reach one second.