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FIGURE 5. Comparison of RBS support values with RAxML SBS, GARLI, and PHYML support values under GTR+I. (a) Superimposec scatterplots for RBS and PHYML BS values for datasets d150, d628, d714, d1481, and d1604 on the respective best-scoring trees. (b) Superimposec scatterplots for RBS and GARLI BS values for datasets d404, d714, d1288, and d1604 on the respective best-scoring trees. (c) Superimposec scatterplots for SBS BS values under GTR+T' and GARLI BS values for datasets d404, d714, d1288, and d1604 on the respective best-scoring trees (d) Superimposed scatterplots for RBS and SBS BS values under GTR+T for datasets d150, d404, d628, d714, d1288, and d1604 on the respectiv: best-scoring trees.

Figure 5 Comparison of RBS support values with RAxML SBS, GARLI, and PHYML support values under GTR+I. (a) Superimposec scatterplots for RBS and PHYML BS values for datasets d150, d628, d714, d1481, and d1604 on the respective best-scoring trees. (b) Superimposec scatterplots for RBS and GARLI BS values for datasets d404, d714, d1288, and d1604 on the respective best-scoring trees. (c) Superimposec scatterplots for SBS BS values under GTR+T' and GARLI BS values for datasets d404, d714, d1288, and d1604 on the respective best-scoring trees (d) Superimposed scatterplots for RBS and SBS BS values under GTR+T for datasets d150, d404, d628, d714, d1288, and d1604 on the respectiv: best-scoring trees.

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FIGURE 1. Outline of lazy subtree rearrangements with cutoff pro- edure.
FIGURE 1. Outline of lazy subtree rearrangements with cutoff pro- edure.
‘IGURE 2. Outline of the RBS (rapid bootstrap) search algorithm and the successive ML search procedure.
‘IGURE 2. Outline of the RBS (rapid bootstrap) search algorithm and the successive ML search procedure.
TABLE 1. Experimental data used and execution time comparison between RBS (rapid bootstrap) and SBS (standard bootstrap). Column #SEQS indicates the number of sequences, #PATT the number of dis- tinct patterns in the alignment that correspond to the length of the likelihood vectors and compute-intensive for-loops, % Gaps the per- centage of completely undetermined character states (e.g., N,O,X,?,- for DNA data) in an ML context, SBS(hrs) indicates the execution time of SBS in hours for 100 BS replicates, RBS(hrs) the execution time for 100 RBS searches, and Speedup the acceleration achieved by RBS.
TABLE 1. Experimental data used and execution time comparison between RBS (rapid bootstrap) and SBS (standard bootstrap). Column #SEQS indicates the number of sequences, #PATT the number of dis- tinct patterns in the alignment that correspond to the length of the likelihood vectors and compute-intensive for-loops, % Gaps the per- centage of completely undetermined character states (e.g., N,O,X,?,- for DNA data) in an ML context, SBS(hrs) indicates the execution time of SBS in hours for 100 BS replicates, RBS(hrs) the execution time for 100 RBS searches, and Speedup the acceleration achieved by RBS.
TABLE 2. BS support value comparison RBS versus SBS. Column #SEQS indicates the dataset, CorrBest denotes the Pearson correlation coefficient p on the best-scoring ML tree, Slope the slope of the linear regression function, Intercept the intercept point of the linear regression function (the worst possible intercept would be +100), CorrAll the Pearson correlation between all bipartitions / splits induced by the BS replicates, Slope the slope of the linear regression function for all splits (the intercept was omitted because it ranged only between —0.013 and + 0.001), SBS(Bips) and RBS(Bips) the total number of bipartitions detected by SBS and RBS, respectively, RF the relative RF-distance (Robinson-Foulds distance) between extended majority rule consensus trees for RBS, and SBS, and finally WRF the analogous weighted RF distance. ITS sequences (d354), fungal sequences (d628, d1908), grasses (d404), as wellas AA sequences (denoted as -AA) of papillomaviruses (d140) and fishes (d775). For two multigene alignments (d140 and d404), we also assessed RBS performance under a partitioned model with joint branch length estimate (denoted as _P). DNA alignments containing 125 up to 7764 sequences. Al though analyses of datasets wi th more than 500 taxa are still relatively uncommon, from an algorithmic point of view, it is bo to th RBS accuracy as well as execu he number of taxa. Eight out of 173 jobs submitted to important to demons trate the scalability of tion times with respect the Vital-IT Web server in February 2008 contained more than 500 taxa, with the largest da he average number of sequences was 125, whereas the average number of seq sequences (t 7" ou of 400 jo our test data f comp ately he computa ments have data. We removed all duplicate sequences as we all entirely undetermined columns For the sake of simplicity, alignments will hencefort taset comprising 5319 uences per alignment bs on the CIPRES We been anonymized to b server amounted to 34). The number of unique alignment site patterns in ranges from 348 to 19,436 and the percent- etely undetermined characters (N,0,X,?,- ta and X,?,-,* for pr 0 79%. The number of alignment columns has deliber- been omitted, because it is irrelevant with respect to ional cost and does no ant measure of alignment size, especially with respect o memory consumption. The taxon names in the align- otein data) from 0.6% represent an impor- hed 1 as from the alignments. h be protect unpublis referenced by the number of taxa, preceded by a “d” as provided in the first column of Table 1. The experi- mental data span a broad range of organisms including rbcL genes (d500, d2554), mammalian sequences (d125, d1288, d2308), bacterial and archaeal sequences (d714, d994, d1481, d1512, d1604, d2000, d4114, d6718, d7764),
TABLE 2. BS support value comparison RBS versus SBS. Column #SEQS indicates the dataset, CorrBest denotes the Pearson correlation coefficient p on the best-scoring ML tree, Slope the slope of the linear regression function, Intercept the intercept point of the linear regression function (the worst possible intercept would be +100), CorrAll the Pearson correlation between all bipartitions / splits induced by the BS replicates, Slope the slope of the linear regression function for all splits (the intercept was omitted because it ranged only between —0.013 and + 0.001), SBS(Bips) and RBS(Bips) the total number of bipartitions detected by SBS and RBS, respectively, RF the relative RF-distance (Robinson-Foulds distance) between extended majority rule consensus trees for RBS, and SBS, and finally WRF the analogous weighted RF distance. ITS sequences (d354), fungal sequences (d628, d1908), grasses (d404), as wellas AA sequences (denoted as -AA) of papillomaviruses (d140) and fishes (d775). For two multigene alignments (d140 and d404), we also assessed RBS performance under a partitioned model with joint branch length estimate (denoted as _P). DNA alignments containing 125 up to 7764 sequences. Al though analyses of datasets wi th more than 500 taxa are still relatively uncommon, from an algorithmic point of view, it is bo to th RBS accuracy as well as execu he number of taxa. Eight out of 173 jobs submitted to important to demons trate the scalability of tion times with respect the Vital-IT Web server in February 2008 contained more than 500 taxa, with the largest da he average number of sequences was 125, whereas the average number of seq sequences (t 7" ou of 400 jo our test data f comp ately he computa ments have data. We removed all duplicate sequences as we all entirely undetermined columns For the sake of simplicity, alignments will hencefort taset comprising 5319 uences per alignment bs on the CIPRES We been anonymized to b server amounted to 34). The number of unique alignment site patterns in ranges from 348 to 19,436 and the percent- etely undetermined characters (N,0,X,?,- ta and X,?,-,* for pr 0 79%. The number of alignment columns has deliber- been omitted, because it is irrelevant with respect to ional cost and does no ant measure of alignment size, especially with respect o memory consumption. The taxon names in the align- otein data) from 0.6% represent an impor- hed 1 as from the alignments. h be protect unpublis referenced by the number of taxa, preceded by a “d” as provided in the first column of Table 1. The experi- mental data span a broad range of organisms including rbcL genes (d500, d2554), mammalian sequences (d125, d1288, d2308), bacterial and archaeal sequences (d714, d994, d1481, d1512, d1604, d2000, d4114, d6718, d7764),
FIGURE 3. Detailed analysis of RBS and SBS support values for dataset d125. (a) Scatterplot for SBS (standard bootstrap) and RBS val- ues drawn on the best-scoring ML tree for d125 (DNA). (b) Scatterplot for SBS and RBS bipartition frequencies for all bipartitions found by SBS or RBS for d125 (DNA). (c) Distribution of SBS and RBS support values on best-scoring ML tree for d125 (DNA).
FIGURE 3. Detailed analysis of RBS and SBS support values for dataset d125. (a) Scatterplot for SBS (standard bootstrap) and RBS val- ues drawn on the best-scoring ML tree for d125 (DNA). (b) Scatterplot for SBS and RBS bipartition frequencies for all bipartitions found by SBS or RBS for d125 (DNA). (c) Distribution of SBS and RBS support values on best-scoring ML tree for d125 (DNA).
FIGURE 4. Scatterplot for SBS and RBS values drawn on the best- scoring ML tree for (a) d140 (AA), (b) d775 (AA), (c) d150 (DNA).
FIGURE 4. Scatterplot for SBS and RBS values drawn on the best- scoring ML tree for (a) d140 (AA), (b) d775 (AA), (c) d150 (DNA).
TABLE 3. Performance of standard and rapid RAxML ML searches and full SBS/RBS ML analyses. Column SLH provides the log likelihood of the best scoring tree found during these 20 searches, whereas column RLH indicates the respective log likelihood obtained by the accelerated ML search. Column SLH(hrs) and RLH(hrs) report the respective ML inference times in hours, Speedup shows the acceleration factor achieved for the ML search, whereas Better? indicates whether the tree obtained via the rapid search had a better log likelihood score than the best tree of the slow searches. Execution times for the combined BS and ML searches are provided in column SF(hrs) for SBS and standard ML, RF(hrs) for RBS and rapid ML. Finally, column SpeedupF provides the overall speedup of RBS over SBS for a full ML analysis, including BS inference and ML search.
TABLE 3. Performance of standard and rapid RAxML ML searches and full SBS/RBS ML analyses. Column SLH provides the log likelihood of the best scoring tree found during these 20 searches, whereas column RLH indicates the respective log likelihood obtained by the accelerated ML search. Column SLH(hrs) and RLH(hrs) report the respective ML inference times in hours, Speedup shows the acceleration factor achieved for the ML search, whereas Better? indicates whether the tree obtained via the rapid search had a better log likelihood score than the best tree of the slow searches. Execution times for the combined BS and ML searches are provided in column SF(hrs) for SBS and standard ML, RF(hrs) for RBS and rapid ML. Finally, column SpeedupF provides the overall speedup of RBS over SBS for a full ML analysis, including BS inference and ML search.
TABLE 4. Computational performance and support value comparison between 100 SBS/RBS and PHYML/GARLI bootstrap analyses. Col: umn PHYML(hrs)/GARLI(hrs) provides the execution time for 100 BS replicates, RF-SBS/RF-RBS the RF distance between SBS/RBS and the espective PHYML/GARLI consensus trees, and WRF-SBS/WRE-RBS the weighted topological distance between SBS/RBS and PHYML/GARL, consensus trees. Finally, column Speedup provides the run time acceleration achieved by RBS over PHYML and GARLI, respectively. We alsc ndicate the average support values for SBS Avg(S), RBS Avg(R), PHYML Avg(P), and GARLI Avg(G).
TABLE 4. Computational performance and support value comparison between 100 SBS/RBS and PHYML/GARLI bootstrap analyses. Col: umn PHYML(hrs)/GARLI(hrs) provides the execution time for 100 BS replicates, RF-SBS/RF-RBS the RF distance between SBS/RBS and the espective PHYML/GARLI consensus trees, and WRF-SBS/WRE-RBS the weighted topological distance between SBS/RBS and PHYML/GARL, consensus trees. Finally, column Speedup provides the run time acceleration achieved by RBS over PHYML and GARLI, respectively. We alsc ndicate the average support values for SBS Avg(S), RBS Avg(R), PHYML Avg(P), and GARLI Avg(G).
TABLE 5. Support value comparison between 100 SBS/RBS and PHYML/GARLI bootstrap analyses. Column CBest-SBS/CBest-RBS provides the correlation of SBS/RBS and PHYML/GARLI support val- ues on the best-scoring ML tree, column CBest-SBS(GAMMA) the cor- relation of SBS support values under GTR+I° with PHYML and GARLI on the best-scoring ML tree. Finally, in columns Slope and Intercept we report the slope and intercept of the linear regression function between SBS under GTR+I and PHYML/GARLI respectively. and GARLI (Stamatakis et al., 2005a; Stamatakis, 2006b; Morrison, 2007). In addition, the average support ob- tained by PHYML (column AVG(P) in Table 4) is sig- nificantly lower than the values obtained via RAxML and GARLI. The data in Table 5 (columns CBest-RBS, CBest-SBS, CBest-SBS(GAMMA)) indicate that the usage of GTR+CAT or GTR+T in RAxML has no notable ef- fect on the correlation between RAxML and PHYML support values; i.e., the differences in the degree of ex- haustiveness of the respective search algorithms have a higher impact on BS values than model details. GARLI and RAxML yield qualitatively comparable trees on medium-sized alignments up to 1000 to 1500 taxa. The only datasets where the correlation between GARLI and SBS/RBS support values is below 0.90 has more than 1500 sequences. The usage of SBS under GTR+T yields a slightly better correlation for d1288 (column CBest-SBS(GAMMA) in Table 5), whereas it is slightly worse than either SBS under GITR+CAT or RBS on datasets d404, d714, d1604. An important result from
TABLE 5. Support value comparison between 100 SBS/RBS and PHYML/GARLI bootstrap analyses. Column CBest-SBS/CBest-RBS provides the correlation of SBS/RBS and PHYML/GARLI support val- ues on the best-scoring ML tree, column CBest-SBS(GAMMA) the cor- relation of SBS support values under GTR+I° with PHYML and GARLI on the best-scoring ML tree. Finally, in columns Slope and Intercept we report the slope and intercept of the linear regression function between SBS under GTR+I and PHYML/GARLI respectively. and GARLI (Stamatakis et al., 2005a; Stamatakis, 2006b; Morrison, 2007). In addition, the average support ob- tained by PHYML (column AVG(P) in Table 4) is sig- nificantly lower than the values obtained via RAxML and GARLI. The data in Table 5 (columns CBest-RBS, CBest-SBS, CBest-SBS(GAMMA)) indicate that the usage of GTR+CAT or GTR+T in RAxML has no notable ef- fect on the correlation between RAxML and PHYML support values; i.e., the differences in the degree of ex- haustiveness of the respective search algorithms have a higher impact on BS values than model details. GARLI and RAxML yield qualitatively comparable trees on medium-sized alignments up to 1000 to 1500 taxa. The only datasets where the correlation between GARLI and SBS/RBS support values is below 0.90 has more than 1500 sequences. The usage of SBS under GTR+T yields a slightly better correlation for d1288 (column CBest-SBS(GAMMA) in Table 5), whereas it is slightly worse than either SBS under GITR+CAT or RBS on datasets d404, d714, d1604. An important result from
FIGURE 6. Inference times for 100 BS replicates with RAxML(SBS), RAxML(RBS), PHYML, and GARLI on DNA datasets; times scaled linearly to 1000 distinct alignment patterns for easier comparability. Note the log-scale on the y-axis.
FIGURE 6. Inference times for 100 BS replicates with RAxML(SBS), RAxML(RBS), PHYML, and GARLI on DNA datasets; times scaled linearly to 1000 distinct alignment patterns for easier comparability. Note the log-scale on the y-axis.
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