  By utilizing the perfectly matched layer (PML) and source transfer
techniques, the diagonal sweeping domain decomposition method (DDM) was
recently developed for solving the high-frequency Helmholtz equation in
$\mathbb{R}^n$, which uses $2^n$ sweeps along respective diagonal directions
with checkerboard domain decomposition. Although this diagonal sweeping DDM is
essentially multiplicative, it is highly suitable for parallel computing of the
Helmholtz problem with multiple right-hand sides when combined with the
pipeline processing since the number of sequential steps in each sweep is much
smaller than the number of subdomains. In this paper, we propose and analyze a
trace transfer-based diagonal sweeping DDM. A major advantage of changing from
source transfer to trace transfer for information passing between neighbor
subdomains is that the resulting diagonal sweeps become easier to analyze and
implement and more efficient, since the transferred traces have only $2n$
cardinal directions between neighbor subdomains while the transferred sources
come from a total of $3^n-1$ cardinal and corner directions. We rigorously
prove that the proposed diagonal sweeping DDM not only gives the exact solution
of the global PML problem in the constant medium case but also does it with at
most one extra round of diagonal sweeps in the two-layered media case, which
lays down the theoretical foundation of the method. Performance and parallel
scalability of the proposed DDM as direct solver or preconditioner are also
numerically demonstrated through extensive experiments in two and three
dimensions.
