Transitive Closure with ZCQ

[Code--Upper Triangular Transitive Closure]
To transitively close a matrix in its ZCQ-representation, we first recurse on its diagonal quadrants, as suggested by (*), to obtain $A^{*}$ and $C^{*}$. We then compute $A^{*}BC^{*}$ with two matrix multiplications. Matrix multiplication in ZCQ is given by the quadrant-based recursive formulation:

$$\left( \begin{array}{ll} A & B\\ C & D \end{array} \righ... ... AE + BG & AF + BH\\ CE + DG & CF + DH \end{array} \right)$$

Summation in ZCQ is given by a coordinate-wise min operation. In addition to the size-one base cases, the efficient implementation of ZCQ includes in both summation and multiplication a sparse case, in which the value of $Z(M)$ is checked first against the dimensions of the submatrices being multiplied. In this context, the base case of multiplication thus always returns 0 (indicating a non-zero element).