  This paper introduces a directional multiscale algorithm for the two
dimensional $N$-body problem of the Helmholtz kernel with applications to high
frequency scattering. The algorithm follows the approach in [Engquist and Ying,
SIAM Journal on Scientific Computing, 29 (4), 2007] where the three dimensional
case was studied. The main observation is that, for two regions that follow a
directional parabolic geometric configuration, the interaction between the
points in these two regions through the Helmholtz kernel is approximately low
rank. We propose an improved randomized procedure for generating the low rank
representations. Based on these representations, we organize the computation of
the far field interaction in a multidirectional and multiscale way to achieve
maximum efficiency. The proposed algorithm is accurate and has the optimal
$O(N\log N)$ complexity for problems from two dimensional scattering
applications. We present numerical results for several test examples to
illustrate the algorithm and its application to two dimensional high frequency
scattering problems.
