## 5 Experimental Results
### Inference Rate Maximization  
Table 5 shows the results for the inference rate maximization objective function for the approaches detailed in Section 4 that are compared to DN\({}^{2}\)PCIoT. Table 6 shows the results for the inference rate maximization objective function for DN\({}^{2}\)PCIoT 30R and DN\({}^{2}\)PCIoT after all the approaches in Table 5. It is worth noting that both Tables 5 and 6 present normalized results, i.e., these results are normalized by the maximum inference rate achieved in each experiment. For instance, in the free-input two-device experiments, considering both Tables 5 and 6, the maximum inference rate was achieved by DN\({}^{2}\)PCIoT after METIS with LeNet 2:1. We take this value and divide it by each result of the free-input two-device experiments. Thus, we have a value of 1.0 for the maximum inference rate in the DN\({}^{2}\)PCIoT after METIS with LeNet 2:1 and the values of the other  
approaches reflect how many times the inference rate was worse than the maximum inference rate. In the first column of both tables, the number indicates the maximum number of devices allowed in each experiment, \"free\" refers to the free-input-layer experiments, and \"locked\" refers to the locked-input-layer experiments. For each experiment in the first column of both tables, there is a range of colors in which the red color represents the worst results while the green color represents the best results. Intermediate results are represented by yellow. As discussed in Section 4, some approaches were not able to produce valid partitionings and this is represented by an \"x\". For LeNet 1:1, as it is a large graph with 2343 vertices, we had to interrupt some executions and we report the best value found followed by an asterisk (\"<<\"). As general results, it is possible to see in Tables 5 and 6 that DN\({}^{2}\)PCIoT 30R and DN\({}^{2}\)PCIoT after approaches led to the best values for all the experiments. DN\({}^{2}\)PCIoT 30R produced results that range from intermediate to the best results, with only 20% of the experiments yielding intermediate results. The DN\({}^{2}\)PCIoT 30R results show the robustness of DN\({}^{2}\)PCIoT, which can achieve reasonable results even when starting from random partitionings. There are some important conclusions that we can draw from Table 5. The per-layer partitioning was the most limiting approach when considering constrained devices because it could only partition the model for the least constrained device setup, which used two devices. It is worth noting that this approach is the one offered by popular ML tools such as TensorFlow, DIANNE, and DeepX. Thus, we show that these tools are not able to execute DNNs in a distributed way in very constrained setups. Moreover, the per-layer partitioning produced suboptimal results for the only setup that it \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & & & & \multicolumn{6}{c}{DN\({}^{2}\)PCIoT after} \\
**Setup** & **30R** & **per** & **per** & **Layers** & **Greedy** & **iRgreedy** & **iRgreedy** & **iRgreedy** & **METIS** & **METIS** \\  & **2:1** & **2:1** & **1:1** & **2:1** & **1:1** & **2:1** & **1:1** & **2:1** & **1:1** \\ \hline
2 free & 1.13 & 1.20 & 1.38 & 1.06 & 1.37 & 1.02 & 1.18 & 1.00 & 1.01 \\
4 free & 1.38 & x & x & 1.16 & 1.25 & 1.16 & 1.14 & 1.00 & 1.01 \\
11 free & 1.19 & x & x & 1.34 & 1.42\({}^{\star}\) & 1.18 & 1.09 & 1.04 & 1.00 \\
56 free & 1.12 & x & x & 2.62 & 5.14\({}^{\star}\) & 2.12 & 1.00\({}^{\star}\) & x & x \\
63 free & 1.00 & x & x & 2.59 & 5.84\({}^{\star}\) & 2.71 & 1.21\({}^{\star}\) & x & x \\ \hline
2 locked & 1.00 & 1.09 & 1.08 & 1.01 & 1.12 & 1.12 & 1.11 & 1.02 & 1.02 \\
4 locked & 1.27 & x & x & 1.29 & 1.25 & 1.00 & 1.45 & 1.25 & 1.23 \\
11 locked & 1.29 & x & x & 1.26 & 1.18 & 1.00 & 1.01 & 1.10 & 1.22 \\
56 locked & 1.46 & x & x & 2.50 & 4.07\({}^{\star}\) & 1.91 & 1.00\({}^{\star}\) & x & x \\
63 locked & 1.17 & x & x & 2.17 & 3.41\({}^{\star}\) & 2.14 & 1.00\({}^{\star}\) & x & x \\ \hline \hline \end{tabular}
\end{table}