Upper and lower bounds on the OBDD-width of a special integer multiplication

Abstract: We consider the Boolean function $${\rm SMul}<em>{n-1}<sup>n(\boldsymbol{x},\boldsymbol{y})$$, which computes the middle bit of the multiplication of two natural numbers represented as $n$-bit binary strings $\boldsymbol{x}$ and $\boldsymbol{y}$, drawn from a restricted domain. We investigate the width of OBDDs computing ${\rm SMul}</em>{n-1}<sup>n$. We introduce a combinatorially defined function $s_<em>(n)$ and show that the width of such OBDDs is $Θ(2^{s_</em>(n)})$.