Figure 14From: Reducing the worst case running times of a family of RNA and CFG problems, using Valiant’s approachThe horizontal 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 horizontally to A I 1 , J and A I 2 , J , where A I 1 , J is computed recursively. (c) If μi, jis of the form ⊕q∈[0,i)(βq, i⊗ αq, j), then the pre-condition for computing A I 2 , J is established, by updating A I 2 , J to be A I 2 , J ⊕ ( β L , I 2 ) T ⊗ A L , J (in this example L = I1, since I2 ends before J starts. The corresponding sub-matrix of β is not shown). In the case where μi, j= ⊕q∈(j,n](αi, q⊗ βj, q), the pre-condition is met without any need for performing a matrix multiplication. Then, A I 2 , J is computed recursively (not shown).Back to article page