Figure 6

The recursion tree. Each node in the tree shows the state of the matrix B when the respective call to Compute-Inside-Sub-Matrix starts. The dotted cells are those that are computed during the call. Black and gray cells are cells whose values were already computed (black cells correspond to empty substrings). The algorithm starts by calling the recursive procedure over the complete matrix. Each visited sub-matrix is decomposed into two halves, which are computed recursively. The recursion visits the sub-matrices according to a pre-order scan on the tree depicted in the figure. Once the first among a pair of sibling recursive calls was concluded, the algorithm uses the new computed portion of data as an input to a fast matrix multiplication subroutine, which facilitate the computation of the second sibling.