  This paper analyses the data rate achieved by various relay selection schemes
in a single-user multi-hop relay network with decode-and-forward (DF) relaying.
While the single-user relay selection problem is well studied in the
literature, research on achievable rate maximization is limited to dual-hop
networks and multi-hop networks with a single relay per hop. We fill this
important gap by focusing on achievable rate maximization in multi-hop,
multi-relay networks. First, we consider optimal relay selection and obtain two
approximations to the achievable rate. Next, we consider three existing
sub-optimal relay selection strategies namely hop-by-hop, ad-hoc and
block-by-block relay selection and obtain exact expressions for the achievable
rate under each of these strategies. We also extend the sliding window based
relay selection to the DF relay network and derive an approximation to the
achievable rate. Further, we investigate the impact of window size in sliding
window based relay selection and show that a window size of three is sufficient
to achieve most of the possible performance gains. Finally, we extend this
analysis to a noise limited multi-user network where the number of available
relay nodes is large compared to the number of users and derive approximations
to the achievable sum-rate.
