Part III. DSHEM and Evaluation 39
Chapter 8. Experimental Analysis of DSHEM 81
8.2. First Experiments on Small Synthetic Graphs
8.2.2. Analysis of Results
Table 8.1: DSHEM parameters for the first set of small synthetic 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
8.2.2. 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 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
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.
Edge cut
DSHEM (Evaluating) versus SHEM (Reference)
Figure 8.1. DSHEM vs. SHEM: effect of -maxvwtm on the edge cut with synthetic 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 8.2. DSHEM vs. SHEM: effect of -maxvwtm on the communication volume with synthetic graphs. Partitioning objective: edge cut on the left, communication volume on the right.
From Figure 8.1, Figure 8.2 and Figure 8.3, it is possible to deduce a pattern on the role of the multiplier -maxvwtm; it is more common to obtain better results by reducing its value. It is also evident that the type of graph has a great influence too, being the 3D square graph (sm3d100p) and the 3D triangular square graph (tsm3d100p) with the highest improvements, and the 2D triangular square graph (tsm2d100p) with the worst results.
Maximum communication volume of all subdomains DSHEM (Evaluating) versus SHEM (Reference)
Figure 8.3. DSHEM vs. SHEM: effect of -maxvwtm on the maximum communication volume of all subdomains with synthetic graphs. Partitioning objective: edge cut on the left, communication volume on the right.
Percentages -dshem_p1, -dshem_p2 and -dshem_p3
Percentages -dshem_p1, -dshem_p2, and -dshem_p3 are used to modify the behavior of the cost function in DSHEM. It may improve the results by selecting the right values according to the type of graph to partition.
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Maximum communication volume of all subdomains DSHEM (Evaluating) versus SHEM (Reference)
Figure 8.4. DSHEM vs. SHEM: effect of -dshem_p1 on the maximum communication volume of all subdomains 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 8.5. DSHEM vs. SHEM: effect of -dshem_p2 on the maximum communication volume of all subdomains 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 8.6. DSHEM vs. SHEM: effect of -dshem_p3 on the maximum communication volume of all subdomains with synthetic graphs. Partitioning objective: edge cut on the left, communication volume on the right.
Percentage -dshem_p1 seems to have no influence based on the results from Figure 8.4. This
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8.2. First Experiments on Small Synthetic Graphs
85 comportment is present for all metrics and all graphs. Percentage -dshem_p3 has a similar output as shown in Figure 8.6. However, Figure 8.5 shows that percentage -dshem_p2 changes the quality of the partitions generated by DSHEM; some types of graphs present a bigger impact by this percentage. Based on the results, it is possible to conclude that lower values for -dshem_p2 produce better results.
After a deeper analysis of the execution of DSHEM, it was found that the conditional, which includes the three percentages, only evaluates to 𝑈𝑅𝑈𝐸 for -dshem_p1 and -dshem_p3 the first time it is executed. The rest of the execution, only -dshem_p2 may evaluate to 𝑈𝑅𝑈𝐸 according to the conditions of the current matching vertices.
Graph Irregularity
The performance of DSHEM is also studied with irregular graphs. Sizes 𝑛 to 𝑒, in Figure 7.2, Figure 7.3 and Figure 7.4, represent the five biggest 2D and all 3D graphs of the set.
Edge cut
DSHEM (Evaluating) versus SHEM (Reference)
Figure 8.7. DSHEM vs. SHEM: effect of irregularity on the edge cut with synthetic graphs and communication volume as partitioning objective.
Communication volume
DSHEM (Evaluating) versus SHEM (Reference)
Figure 8.8. DSHEM vs. SHEM: effect of irregularity on the communication volume with synthetic graphs and communication volume as partitioning objective.
From the results, it is possible to see that some types of graphs have greater impact than others. One of the most stable types is the 2D dense triangular square graph (dtsm2d) when evaluating the edge cut.
While others may show some spikes, there is not a clear pattern. The introduced regularity in the graphs
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Graph sizes and percentage of edges
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has an impact, but the percentage of irregularity is not necessarily proportional to the impact on the quality of the partition.
Regarding 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.
Maximum communication volume of all subdomains DSHEM (Evaluating) versus SHEM (Reference)
Figure 8.9. DSHEM vs. SHEM: effect of irregularity on the maximum communication volume of all subdomains with synthetic graphs and communication volume as partitioning objective.
Refinement
METIS is executed without refinement for both, SHEM and DSHEM, to analyze the effect on the partitioning process. Whether the partitioning objective is the edge cut or the communication volume, the results remain the same; SHEM and DSHEM perform the matching without a partitioning objective and without the refinement process no objective is optimized.
Edge cut
DSHEM (Evaluating) versus SHEM (Reference)
Figure 8.10. DSHEM vs. SHEM: effect of refinement on the edge cut with synthetic graphs. Partitioning objective: edge cut on the left, communication volume on the right.
Figure 8.10 shows the effect of the refinement when the edge cut is evaluated. It is clear that the 2D and 3D square graphs (sm2d100p and sm3d100p) benefit from -dshem_p2 with values from to 100. The 3D triangular square graph (tsm3d100p) has a similar pattern, however in the negative side of the plot.
The rest of the graphs have a more stable behavior.
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