## 5 Computational Results
### Results for Extended SCOPCath Benchmark  
Our exact \(k\)-NN classification can also be successfully applied to larger benchmarks, like extended SCOPCath, which are more representative of databases, such as SCOP. Here, the benefit of using a metric distance, triangle inequality and \(k\)-NN classification is more pronounced. Remarkably, our classification run time on this benchmark that is about an order of magnitude larger than SCOPCath is for most queries of the same order of magnitude as run times on SCOPCath (except for some queries that need an extremely long run time and finally cannot be assigned exactly). Furthermore, here, run time varies extremely between queries, between \(0.15\) and \(85.63\) h for queries of the four major classes that could be assigned exactly. The median run time for all \(1120\) exactly assigned extended SCOPCath queries is \(3.8\) h. The classification results for extended SCOPCath are shown in Table 3. Slightly more queries have been assigned correctly compared to SCOPCath, and significantly more queries have been assigned exactly. Both may reflect that there are now more similar structures within the targets. Further, the number of ties is decreased.  
Figure 6: Boxplots of the percentage of removed targets at each iteration during triangle and pairwise dominance for the \(236\) queries of the SCOPCath benchmark.  
Figure 7 displays the progress of the computation. Here, many more target structures are removed by triangle dominance and within the very first iteration of pairwise dominance compared to the SCOPCath benchmark. For example, for most queries, more than 60% of targets are removed by triangle dominance alone. Only very few queries need to explicitly compute the distance to a large percentage of the targets, and almost \(75\)% of queries can be assigned after only one round of pairwise dominance.  
\begin{table}
\begin{tabular}{l r r r r r r r r r r} \hline
**k** & **10** & **9** & **8** & **7** & **6** & **5** & **4** & **3** & **2** & **1** \\ \hline \# correct & 1303 & 1331 & 1334 & 1341 & 1341 & 1346 & 1344 & 1351 & 1348 & 1361 \\ \# exact & 1120 & 1182 & 1228 & 1271 & 1286 & 1339 & 1341 & 1352 & 1347 & 1368 \\ \# exact and correct & 1104 & 1166 & 1215 & 1257 & 1276 & 1329 & 1330 & 1341 & 1343 & 1360 \\ \# ties & 35 & 5 & 12 & 6 & 11 & 7 & 9 & 3 & 17 & 0 \\ \# TM-align correct & 1311 & 1347 & 1346 & 1350 & 1351 & 1354 & 1352 & 1353 & 1351 & 1361 \\ \# TM-align ties & 39 & 4 & 7 & 4 & 6 & 4 & 4 & 5 & 15 & 0 \\ \hline \end{tabular}
\end{table}
Table 3: Classification results showing the number of queries out of the overall \(1369\) queries that have been assigned to a superfamily, the number of correct assignments, the number of assignments computed exactly, thereof the number of correct classifications and the number of ties that do not allow a superfamily assignment based on majority vote. The last two lines display the number of correct assignments and ties for \(k\)-NN classification using TM-align.  
Figure 7: Boxplots of the percentage of removed targets at each iteration during triangle and pairwise dominance for the \(1369\) queries of the extended SCOPCath benchmark.