# Table 1 Time complexities of several VMT problems

Problem Standard DP running time Implicit [explicit] VMT algorithm running time
Results previously published CFG Recognition/Parsing $\Theta \left({n}^{3}\right)$)  $\Theta \left(DB\left(n\right)\right)\left[\Theta \left({n}^{2.38}\right)\right]$
WCFG Parsing $\Theta \left({n}^{3}\right)$ $\Theta \left(MP\left(n\right)\right)\left[\stackrel{˜}{O}\left(\frac{{n}^{3}}{{\mathrm{log}}^{2}n}\right)\right]$
RNA Single Strand Folding $\Theta \left({n}^{3}\right)$[6, 7] $\Theta \left(MP\left(n\right)\right)\left[\stackrel{˜}{O}\left(\frac{{n}^{3}}{{\mathrm{log}}^{2}n}\right)\right]$
RNA Partition Function $\Theta \left({n}^{3}\right)$ $\Theta \left(MP\left(n\right)\right)\left[\Theta \left({n}^{2.38}\right)\right]$
In this paper WCFG Inside-Outside $\Theta \left({n}^{3}\right)$ $\Theta \left(DB\left(n\right)\right)\left[\Theta \left({n}^{2.38}\right)\right]$
RNA Base Pair Binding Probabilities $\Theta \left({n}^{3}\right)$ $\Theta \left(DB\left(n\right)\right)\left[\Theta \left({n}^{2.38}\right)\right]$
RNA Simultaneous Alignment and Folding $\Theta \left({\left(n/2\right)}^{{}^{3m}}\right)$ $\Theta \left(MP\left({n}^{m}\right)\right)\left[\stackrel{˜}{O}\left(\frac{{n}^{3}m}{m{\mathrm{log}}^{2}n}\right)\right]$
RNA-RNA Interaction $\Theta \left({n}^{6}\right)$ $\Theta \left(MP\left({n}^{2}\right)\right)\left[\stackrel{˜}{O}\left(\frac{{n}^{6}}{{\mathrm{log}}^{2}n}\right)\right]$
RNA-RNA Interaction Partition Function $\Theta \left({n}^{6}\right)$ $\Theta \left(DB\left(n\right)\right)\left[\Theta \left({n}^{4.75}\right)\right]$
RNA Sequence to Structured-Sequence Alignment $\Theta \left({n}^{4}\right)$ $\Theta \left(nMP\left(n\right)\right)\left[\stackrel{˜}{O}\left(\frac{{n}^{4}}{{\mathrm{log}}^{2}n}\right)\right]$
1. The notation $\stackrel{~}{O}$ corresponds to the standard big-O notation, hiding some polylogarithmic factors. 