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Table 1 Time complexities of several VMT problems

From: Reducing the worst case running times of a family of RNA and CFG problems, using Valiant’s approach

  Problem Standard DP running time Implicit [explicit] VMT algorithm running time
Results previously published CFG Recognition/Parsing Θ( n 3 )) [2022] Θ(DB(n)) [ Θ ( n 2.38 ) ] [23]
  WCFG Parsing Θ( n 3 )[17] Θ ( M P ( n ) ) [ O ˜ ( n 3 log 2 n ) ] [25]
  RNA Single Strand Folding Θ( n 3 )[6, 7] Θ ( M P ( n ) ) [ O ˜ ( n 3 log 2 n ) ] [25]
  RNA Partition Function Θ( n 3 )[10] Θ(MP(n)) [ Θ ( n 2.38 ) ] [25]
In this paper WCFG Inside-Outside Θ( n 3 )[43] Θ(DB(n)) [ Θ ( n 2.38 ) ]
  RNA Base Pair Binding Probabilities Θ( n 3 )[10] Θ(DB(n)) [ Θ ( n 2.38 ) ]
  RNA Simultaneous Alignment and Folding Θ ( ( n / 2 ) 3 m ) [15] Θ ( M P ( n m ) ) [ O ˜ ( n 3 m m log 2 n ) ]
  RNA-RNA Interaction Θ( n 6 )[9] Θ ( M P ( n 2 ) ) [ O ˜ ( n 6 log 2 n ) ]
  RNA-RNA Interaction Partition Function Θ( n 6 )[12] Θ(DB(n)) [ Θ ( n 4.75 ) ]
  RNA Sequence to Structured-Sequence Alignment Θ( n 4 )[13] Θ ( n M P ( n ) ) [ O ˜ ( n 4 log 2 n ) ]
  1. The notation O ~ corresponds to the standard big-O notation, hiding some polylogarithmic factors.