  In this work, we propose three pilot assignment schemes to reduce the effect
of pilot contamination in cell-free massive multiple-input-multiple-output
(MIMO) systems. Our first algorithm, which is based on the idea of random
sequential adsorption (RSA) process from the statistical physics literature,
can be implemented in a distributed and scalable manner while ensuring a
minimum distance among the co-pilot users. Further, leveraging the rich
literature of the RSA process, we present an approximate analytical approach to
accurately determine the density of the co-pilot users as well as the pilot
assignment probability for the typical user in this network. We also develop
two optimization-based centralized pilot allocation schemes with the primary
goal of benchmarking the RSA-based scheme. The first centralized scheme is
based only on the user locations (just like the RSA-based scheme) and
partitions the users into sets of co-pilot users such that the minimum distance
between two users in a partition is maximized. The second centralized scheme
takes both user and remote radio head (RRH) locations into account and provides
a near-optimal solution in terms of sum-user spectral efficiency (SE). The
general idea is to first cluster the users with similar propagation conditions
with respect to the RRHs using spectral graph theory and then ensure that the
users in each cluster are assigned different pilots using the branch and price
(BnP) algorithm. Our simulation results demonstrate that despite admitting
distributed implementation, the RSA-based scheme has a competitive performance
with respect to the first centralized scheme in all regimes as well as to the
near-optimal second scheme when the density of RRHs is high.
