This paper is concerned with fast solution of high frequency acoustic scattering
problems in two dimensions. We introduce a directional multiscale algorithm for the $N$-body problem
of the two dimensional Helmholtz kernel. The algorithm follows the approach developed in Engquist
and Ying, SIAM J. Sci. Comput., 29 (4), 2007, where the three dimensional case was studied. The
main observation is that, for two regions that follow a directional parabolic geometric conguration,
the interaction between these two regions through the 2D Helmholtz kernel is approximately low rank.
We propose an improved randomized procedure for generating the low rank separated representation
for the interaction between these regions. Based on this representation, the computation of the far field
interaction is organized in a multidirectional and multiscale way to achieve maximum efficiency.
The proposed algorithm is accurate and has the optimal $O(NlogN)$ complexity for problems from
two dimensional scattering applications. Finally, we combine this fast directional algorithm with
standard boundary integral formulations to solve acoustic scattering problems that are of thousands
of wavelengths in size.