  In a computing center with a huge amount of machines, when a job arrives, a
dispatcher need to decide which machine to route this job to based on limited
information. A classical method, called the power-of-$d$ choices algorithm is
to pick $d$ servers independently at random and dispatch the job to the least
loaded server among the $d$ servers. In this paper, we analyze a low-randomness
variant of this dispatching scheme, where $d$ queues are sampled through $d$
independent non-backtracking random walks on a $k$-regular graph $G$. Under
certain assumptions of the graph $G$ we show that under this scheme, the
dynamics of the queuing system converges to the same deterministic ordinary
differential equation (ODE) for the power-of-$d$ choices scheme. We also show
that the system is stable under the proposed scheme, and the stationary
distribution of the system converges to the fixed point of the ODE.
