# 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

$\Theta \left({n}^{3}\right)$) [2022]

$\Theta \left(DB\left(n\right)\right)\left[\Theta \left({n}^{2.38}\right)\right]$[23]

WCFG Parsing

$\Theta \left({n}^{3}\right)$[17]

$\Theta \left(MP\left(n\right)\right)\left[\stackrel{˜}{O}\left(\frac{{n}^{3}}{{\mathrm{log}}^{2}n}\right)\right]$[25]

RNA Single Strand Folding

$\Theta \left({n}^{3}\right)$[6, 7]

$\Theta \left(MP\left(n\right)\right)\left[\stackrel{˜}{O}\left(\frac{{n}^{3}}{{\mathrm{log}}^{2}n}\right)\right]$[25]

RNA Partition Function

$\Theta \left({n}^{3}\right)$[10]

$\Theta \left(MP\left(n\right)\right)\left[\Theta \left({n}^{2.38}\right)\right]$[25]

In this paper

WCFG Inside-Outside

$\Theta \left({n}^{3}\right)$[43]

$\Theta \left(DB\left(n\right)\right)\left[\Theta \left({n}^{2.38}\right)\right]$

RNA Base Pair Binding Probabilities

$\Theta \left({n}^{3}\right)$[10]

$\Theta \left(DB\left(n\right)\right)\left[\Theta \left({n}^{2.38}\right)\right]$

RNA Simultaneous Alignment and Folding

$\Theta \left({\left(n/2\right)}^{{}^{3m}}\right)$[15]

$\Theta \left(MP\left({n}^{m}\right)\right)\left[\stackrel{˜}{O}\left(\frac{{n}^{3}m}{m{\mathrm{log}}^{2}n}\right)\right]$

RNA-RNA Interaction

$\Theta \left({n}^{6}\right)$[9]

$\Theta \left(MP\left({n}^{2}\right)\right)\left[\stackrel{˜}{O}\left(\frac{{n}^{6}}{{\mathrm{log}}^{2}n}\right)\right]$

RNA-RNA Interaction Partition Function

$\Theta \left({n}^{6}\right)$[12]

$\Theta \left(DB\left(n\right)\right)\left[\Theta \left({n}^{4.75}\right)\right]$

RNA Sequence to Structured-Sequence Alignment

$\Theta \left({n}^{4}\right)$[13]

$\Theta \left(nMP\left(n\right)\right)\left[\stackrel{˜}{O}\left(\frac{{n}^{4}}{{\mathrm{log}}^{2}n}\right)\right]$

1. The notation $\stackrel{~}{O}$ corresponds to the standard big-O notation, hiding some polylogarithmic factors.