  The \textsc{Arbitrary Pattern Formation} (\textsc{Apf}) is a widely studied
in distributed computing for swarm robots. This problem asks to design a
distributed algorithm that allows a team of identical, autonomous mobile robots
to form any arbitrary pattern given as input. This paper considers that the
robots are operating on a two-dimensional infinite grid. Robots are initially
positioned on distinct grid points forming an asymmetric configuration (no two
robots have the same snapshot). They operate under a fully asynchronous
scheduler and do not have any access to a global coordinate system, but they
will align the axes of their local coordinate systems along the grid lines. The
previous work dealing with \textsc{Apf} problem solved it in
$O(\mathcal{D}^2k)$ robot movements under similar conditions, where
$\mathcal{D}$ is the side of the smallest square that can contain both initial
and target configuration and, $k$ is the number of robots. Let
$\mathcal{D}'=\max\{\mathcal{D},k\}$. This paper presents two algorithms of
\textsc{Apf} on an infinite grid. The first algorithm solves the \textsc{Apf}
problem using $O(\mathcal{D}')$ asymptotically move optimal. The second
algorithm solves the \textsc{Apf} problem in $O(\mathcal{D}')$ epochs, which we
show is asymptotically optimal.

Distributed, Parallel, and Cluster Computing