Table 1 Time complexities of several VMT problems

Results previously published

CFG Recognition/Parsing

$\Theta (n^3)$) [20–22]

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

WCFG Parsing

$\Theta (n^3)$[17]

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

RNA Single Strand Folding

$\Theta (n^3)$[6, 7]

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

RNA Partition Function

$\Theta (n^3)$[10]

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

In this paper

WCFG Inside-Outside

$\Theta (n^3)$[43]

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

RNA Base Pair Binding Probabilities

$\Theta (n^3)$[10]

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

RNA Simultaneous Alignment and Folding

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

$\Theta (MP(n^m))[\tilde{O}(\frac{n^{3}m}{m{\log }^{2}n})]$

RNA-RNA Interaction

$\Theta (n^6)$[9]

$\Theta (MP(n^2))[\tilde{O}(\frac{n^6}{{\log }^{2}n})]$

RNA-RNA Interaction Partition Function

$\Theta (n^6)$[12]

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

RNA Sequence to Structured-Sequence Alignment

$\Theta (n^4)$[13]

$\Theta (nMP(n))[\tilde{O}(\frac{n^4}{{\log }^{2}n})]$