Figure 13From: Reducing the worst case running times of a family of RNA and CFG problems, using Valiant’s approachThe vertical decomposition in the Outside VMT algorithm, for the case where | I | ≤ | J |. (a) AI, Jsustains the pre-condition (see Figure 12 for notations). (b) AI, Jis partitioned vertically to A I , J 1 and A I , J 2 , where A I , J 1 is computed recursively. (c) If μi, jis of the form ⊕ q∈(j,n](αi, q⊗ βj, q), then the pre-condition for computing A I , J 2 is established, by updating A I , J 2 to be A I , L ⊗ ( β J 2 , L ) T ⊕ A I , J 2 (in this example L = J1, since I ends before J1 starts. The corresponding sub-matrix of β is not shown). In the case where μi, j= ⊕q∈[0,i)(βq, i⊗ αq, j), the pre-condition is met without any need for performing a matrix multiplication. Then, A I , J 2 is computed recursively (not shown).Back to article page