## 3 Benchmarking
### Complexity Analysis  
Figure 6c shows a heat map of the median running time of \(\text{LUMAWIG}_{\text{R}}\) over 100 computations per pixel of the bottleneck distance between pairs of persistence diagram with varying number of points: for \(i\leq j\), pixel \((i,j)\) represents the median running time for the computation of the bottleneck distance between a diagram with \(i\) thousands of points and another diagram with \(j\) thousands of points, such that each set of points has death times uniformly chosen from the interval range \((0,2000i)\) and \((0,2000j)\) respectively. It can be inferred from this figure that the best running times happen along the main diagonal, as well as the upper and left portions of the heat map. These correspond to two specific cases: when the diagrams have equal number of points, or when one diagram has \begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline  & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \(\text{LUMAWIG}_{\text{R}}\) & 0.230 & 0.651 & 1.203 & 1.971 & 2.744 & 3.957 & 5.003 & 6.813 & 8.348 & 9.983 \\ \(\text{LUMAWIG}_{\text{PY}}\) & 0.231 & 0.659 & 1.221 & 2.004 & 2.797 & 4.037 & 5.107 & 6.928 & 8.498 & 10.181 \\ \hline \(\text{LUMAWIG}_{\text{R}}\) & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\ \hline \(\text{LUMAWIG}_{\text{R}}\) & 11.029 & 12.227 & 14.985 & 15.733 & 18.983 & 21.588 & 23.580 & 26.801 & 29.425 & 33.316 \\ \(\text{LUMAWIG}_{\text{PY}}\) & 11.255 & 12.483 & 15.296 & 16.078 & 19.410 & 22.087 & 24.124 & 27.438 & 30.153 & 34.129 \\ \hline \(\text{LUMAWIG}_{\text{R}}\) & 21 & 22 & 23 & 24 & 25 & 26 & 27 & 28 & 29 & 30 \\ \hline \(\text{LUMAWIG}_{\text{R}}\) & 38.555 & 39.818 & 41.080 & 44.324 & 49.933 & 54.441 & 57.183 & 60.196 & 66.510 & 72.948 \\ \(\text{LUMAWIG}_{\text{PY}}\) & 39.427 & 40.734 & 42.073 & 45.407 & 51.129 & 55.725 & 58.517 & 61.605 & 68.200 & 74.879 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of significant decrease (at confidence level \(\alpha=0.95\)) in running time (in seconds) for paired tests versus Hera. Column labels are in thousands of points. Figure 6: (a,b) Boxplots of relative differences between the bottleneck computation outputs of the indicated pair of implementations. (c) Heat map of the median running times of \(\text{LUMAWIG}_{\text{R}}\). Each pixel represents the median running time (in seconds) for 100 computations of dimension zero bottleneck distance between diagrams. The number of points in the diagrams are in units of 1000.  
overwhelmingly more points than the other. In contrast, regions in the heat map that show increased running times correspond to the case when a diagram that has a large number of points is compared to another that has about half as many points. This observation is supported in the next figure. Figure 7 shows several scatter plots with fitted curves of the median running times of Lumawigr over 100 computations of dimension zero bottleneck distance. The label in each scatterplot represents the number of points in the fixed base diagram, and a point in the scatterplot at the \(k\) thousand mark along the horizontal axis represents the median running time over 100 computations of the bottleneck distance between the base diagram and another diagram with \(k\) thousand points whose death times are uniformly chosen from the interval range \((0,2k)\). To further investigate the observations above, we examine the performance of Lumawigr in the computation of dimension zero bottleneck distance in four pairs of settings for size of the diagrams and the range of values the death times are drawn from. The first is when Lumawigr is tasked to compare two persistence diagrams with the same number of points whose death times are drawn  
Figure 7: Scatter plots with fitted curves of the median running times (in seconds) of Lumawigr over 100 computations of dimension zero bottleneck distance between a base diagram with the labeled number of points and a diagram with \(k\) thousands of points, for \(k=1,2,\ldots,100\). from the same range of values.