Table 3 The outside VMT algorithm

   1:    Allocate a matrix A of size ||s|| × ||s||, and initialize all entries in A with ϕ elements.

   2:    Call Compute-Outside-Sub-Matrix ([0, n], [0, n]), where n is the length of s.

   3:    return A

Compute-Outside-Sub-Matrix (I, J)

pre-condition: The values in A_{[0,I], [J,n]}, excluding the values in A _I,J , are computed.

   If μ _i,j = ⊕_q∈[0,i)(β _q,i ⊗ α _q,j ), then A _I,J = (β_[0,I),I)^T ⊗ α_[0,I),J. Else, if μ _i,j = ⊕_q∈(j,n](α _i,q ⊗ β _j,q ), then A _I,J = α_I,(J,n]⊗ (β_J,(J,n])^T.

post-condition: A_{[0,I], [J,n]}= α_{[0,I], [J,n]}.

   1:    if I = [i, i] and J = [j, j] then

   2:       If i ≤ j, compute α _i,j (in o (||s||) running time) by querying computed values in A and the value μ _i,j which is stored in A _i,j . Update A _i,j ← α _i,j .

   3:    else

   4:       if |I| ≤ |J| then

   5:          Let J₁ and J₂ be the two intervals such that J₂J₁ = J, and |J₂| = ⌊|J|/2⌋.

   6:          Call Compute-Outside-Sub-Matrix (I, J₁).

   7:          if μ _i,j is of the form ⊕_q∈(j,n](α _i,q ⊗ β _j,q ) then

   8:             Let L be the interval such that L(J, n] = (J₂, n].

   9:             Update $A_{I,J_2}\leftarrow \left(A_{I,L}\otimes {\left({\beta }_{J_2,L}\right)}^T\right)\oplus A_{I,J_2}$.

   10:          end if

   11:          Call Compute-Outside-Sub-Matrix (I, J₂).

   12:       else

   13:          Let I₁ and I₂ be the two intervals such that I₁I₂ = I, and |I₁| = ⌊|I|/2⌋.

   14:          Call Compute-Outside-Sub-Matrix (I₁, J).

   15:          if μ _i,j is of the form ⊕_q∈[0,i)(β _q,i ⊗ α _q,j ) then

   16:             Let L be the interval such that [0, I)L = [0, I₂).

   17:             Update $A_{I_2,J}\leftarrow A_{I_2,J}\oplus \left({\left({\beta }_{L,I_2}\right)}^T\otimes A_{L,J}\right)$.

   18:          end if

   19:          Call Compute-Outside-Sub-Matrix (I₂, J).

   20:       end if

   21:    end if