Generalizations of the distributed Deutsch-Jozsa promise problem

In the {\em distributed Deutsch-Jozsa promise problem}, two parties are to determine whether their respective strings $x,y\in{0,1}^n$ are at the {\em Hamming distance} $H(x,y)=0$ or $H(x,y)=\frac{n}{2}$. Buhrman et al. (STOC' 98) proved that the exact {\em quantum communication complexity} of this problem is ${\bf O}(\log {n})$ while the {\em deterministic communication complexity} is ${\bf \Omega}(n)$. This was the first impressive (exponential) gap between quantum and classical communication complexity. In this paper, we generalize the above distributed Deutsch-Jozsa promise problem to determine, for any fixed $\frac{n}{2}\leq k\leq n$, whether $H(x,y)=0$ or $H(x,y)= k$, and show that an exponential gap between exact quantum and deterministic communication complexity still holds if $k$ is an even such that $\frac{1}{2}n\leq k<(1-\lambda) n$, where $0< \lambda<\frac{1}{2}$ is given. We also deal with a promise version of the well-known {\em disjointness} problem and show also that for this promise problem there exists an exponential gap between quantum (and also probabilistic) communication complexity and deterministic communication complexity of the promise version of such a disjointness problem. Finally, some applications to quantum, probabilistic and deterministic finite automata of the results obtained are demonstrated.