  We have studied $A+A \rightarrow \emptyset$ reaction-diffusion model on a
ring, with a bias $\epsilon$ $(0 \leq \epsilon \leq 0.5)$ of the random walkers
$A$ to hop towards their nearest neighbor. Though the bias is local in space
and time, we show that it alters the universality class of the problem. The $z$
exponent, which describes the growth of average spacings between the walkers
with time, changes from the value 2 at $\epsilon=0$ to the mean-field value of
unity for any non-zero $\epsilon$. We study the problem analytically using
independent interval approximation and compare the scaling results with that
obtained from simulation. The distribution $P(k,t)$ of the spacing $k$ between
two walkers (per site) is given by $t^{-2/z} f(k/t^{1/z})$ as expected;
however, the scaling function shows different behaviour in the two approaches.
