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8.5. Experiments on Real Life Graphs
This particular set of experiments uses the graphs presented in Table 7.9 of Chapter 7. It is a set of real life graphs chosen to evaluate the performance of DSHEM and compare it with SHEM and Random.
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8.5.1. Execution Parameters
Four main parameters are used to tune up DSHEM, namely -maxvwtm, -dshem_p1, dshem_p2, and -dshem_p3. The values chosen for the fourth set of experiments are presented in Table 8.4. This particular set produces 1715 different combinations of values, giving a wide view of the performance of DSHEM. They are based on the results from previous experimental results, and used to confirm the findings with the synthetic graphs.
Table 8.4: DSHEM parameters for the set of real life graphs.
-maxvwtm -dshem_p1 -dshem_p2 -dshem_p3
140 to 160, step 5 91 to 109, step 3 91 to 109, step 3 91 to 109, step 3
8.5.2. 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. First, the effect of the multiplier -maxvwtm is evaluated. Next, the three percentages -dshem_p1, -dshem_p2, and -dshem_p3 are examined to understand their influence. The values used for this particular set of experiments are based on the results of the previous three sets; they help validate the initial findings. The refinement and its influence on DSHEM are also studied to confirm the previous results obtained from the synthetic graphs. 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
The multiplier -maxvwtm limits the size of vertices during the coarsening process. Reducing its value produces more balanced initial partitions and the refinement process is also optimized. However, a low value may have also undesired effects such as the inability to match vertices that could lead to an infinite loop trying to contract the graph without success.
Based on the results obtained from the experiments with synthetic graphs, the values chosen for the multiplier -maxvwtm are designed to match those of the first set. They provide a wide view and reduce the number of experiments in the set.
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Edge cut
DSHEM (Evaluating) versus SHEM (Reference)
Figure 8.36. DSHEM vs. SHEM: effect of -maxvwtm on the edge cut with real life graphs. Partitioning objective: edge cut on the left, communication volume on the right.
From the results shown in Figure 8.36 with real life graphs, it is possible to confirm the findings with the synthetic counterparts. The graph ef_ocean has a similar geometry to that of the 3D square graph and presents similar behavior: improvement over most of the values of multiplier -maxvwtm. The graphs ef_4elt and ef_sphere have a triangular geometry, as the 2D triangular square graph, and DSHEM produces poor results with them too.
Figure 8.37 and Figure 8.38 present the evaluation of the total communication volume and the maximum communication volume of all subdomains respectively. It is also evident that the graph ef_ocean has a clear improvement with DSHEM. The other two graphs, with triangular geometry, show degradation in the quality of the partition, as initially found with the synthetic graphs.
Communication volume
DSHEM (Evaluating) versus SHEM (Reference)
Figure 8.37. DSHEM vs. SHEM: effect of -maxvwtm on the communication volume with real life graphs. Partitioning objective: edge cut on the left, communication volume on the right.
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Maximum communication volume of all subdomains DSHEM (Evaluating) versus SHEM (Reference)
Figure 8.38. DSHEM vs. SHEM: effect of -maxvwtm on the maximum communication volume of all subdomains with real life graphs. Partitioning objective: edge cut on the left, communication volume on the right.
The results may suggest that smaller values for the multiplier -maxvwtm improve the results. Though, the results with the second set of experiments demonstrate that variations on the quality of the partition depend on the instance of the problem and not the multiplier.
Percentages -dshem_p1, -dshem_p2 and -dshem_p3
Percentages -dshem_p1 and -dshem_p3 are excluded from the analysis as they do not play any role in the partitioning process. Percentage -dshem_p2 is used to modify the behavior of the cost function in DSHEM and improve the partition.
Values inferior to 100 for the percentage -dshem_p2 produce better results with the graph ef_ocean, as shown in Figure 8.39, Figure 8.40 and Figure 8.41. Whether the partitioning objective is the edge cut or communication volume, the graph ef_ocean indisputably presents a benefit from DSHEM. The graphs ef_4elt and ef_sphere remain with little improvement or degradation.
Edge cut
DSHEM (Evaluating) versus SHEM (Reference)
Figure 8.39. DSHEM vs. SHEM: effect of -dshem_p2 on the edge cut with real life graphs. Partitioning objective: edge cut on the left, communication volume on the right.
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Communication volume
DSHEM (Evaluating) versus SHEM (Reference)
Figure 8.40. DSHEM vs. SHEM: effect of -dshem_p2 on the communication volume with real life 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 8.41. DSHEM vs. SHEM: effect of -dshem_p2 on the maximum communication volume of all subdomains with real life graphs. Partitioning objective: edge cut on the left, communication volume on the right.
These results confirm the initial findings with the synthetic graphs that suggest the performance of DSHEM is superior with graphs having a quadrangular-like geometry.
Refinement
METIS is executed without refinement for both, SHEM and DSHEM, to analyze its effect on the partitions. This process is responsible of the majority of the execution time; it improves the initial partition according to the partitioning objective.
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Edge cut
DSHEM (Evaluating) versus SHEM (Reference)
Figure 8.42. DSHEM vs. SHEM: effect of refinement on the edge cut with real life graphs. Partitioning objective: edge cut on the left, communication volume on the right.
Communication volume
DSHEM (Evaluating) versus SHEM (Reference)
Figure 8.43. DSHEM vs. SHEM: effect of refinement on the communication volume with real life graphs. Partitioning objective: edge cut on the left, communication volume on the right.
It is interesting to note that the quality of the partition, when the edge cut is evaluated, improves when values for the percentage -dshem_p2 are lower than 100, see Figure 8.42; it is the opposite with synthetic graphs and square geometry. It is not clear why DSHEM presents this behavior with the graph ef_ocean. The graphs with triangular geometry have a similar output to the synthetic 2D triangular square graph.
Regarding the total communication volume and the maximum communication volume of all subdomains, Figure 9.31 and Figure 9.32 depict a congruent scenario with the synthetic graphs. The total communication volume improves with values for the percentage -dshem_p2 being 100 or higher.
The maximum communication volume of all subdomains is more irregular, but in general, lower values produce better results.
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Maximum communication volume of all subdomains DSHEM (Evaluating) versus SHEM (Reference)
Figure 8.44. DSHEM vs. SHEM: effect of refinement on the maximum communication volume of all subdomains with real life graphs. Partitioning objective: edge cut on the left, communication volume on the right.
Execution Time
The two graphs with triangular geometries confirm that this type keeps the partitioning time balanced between edge cut and communication volume, as depicted in Figure 8.45 and Figure 8.46. There is virtually no difference in execution time whether the edge cut or communication volume is optimized by the refinement process.
Figure 8.45. Partitioning time with graph ef_4elt and greedy refinement.
Figure 8.46. Partitioning time with graph ef_sphere and greedy refinement.
The experimental results confirm that the time spent on the refinement process greatly depends on the number of subdomains; from a few seconds for 2 subdomains to some minutes for 32 subdomains with
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