terms of the channel gains. We find expression for optimal relay location in the single relay placement problem (Theorem \[theorem:single\_relay\_total\_power\]). For the $N$ relay placement problem, numerical study shows that, the relay nodes are clustered near the source at low attenuation and are placed uniformly at high attenuation. Theorem \[theorem:large\_nodes\_uniform\] shows that, by placing large number of relays uniformly, we can achieve a rate arbitrarily close to the AWGN capacity. Only this part of our current paper was published in the conference version [@chattopadhyay-etal12optimal-capacity-relay-placement-line].

-   [**Optimal As-You-Go Deployment:**]{} In Section \[sec:mdp\_total\_power\], we consider the problem of placing relay nodes in a deploy-as-you-go manner, so as to connect a source and a sink separated by an unknown distance, modeled as an exponentially distributed random variable $L$. Specifically, the problem is to start from a source, and walk along a line, placing relay nodes as we go, until the line ends, at which point the sink is placed. With a sum power constraint, the aim is to maximize a capacity limiting term derived from the deployment problem for known $L$, while constraining the expected number of relays. We “relax” the expected number of relays constraint via a Lagrange multiplier, and formulate the 