  We consider the N-relay Gaussian diamond network where a source node
communicates to a destination node via N parallel relays through a cascade of a
Gaussian broadcast (BC) and a multiple access (MAC) channel. Introduced in 2000
by Schein and Gallager, the capacity of this relay network is unknown in
general. The best currently available capacity approximation, independent of
the coefficients and the SNR's of the constituent channels, is within an
additive gap of 1.3 N bits, which follows from the recent capacity
approximations for general Gaussian relay networks with arbitrary topology.
  In this paper, we approximate the capacity of this network within 2 log N
bits. We show that two strategies can be used to achieve the
information-theoretic cutset upper bound on the capacity of the network up to
an additive gap of O(log N) bits, independent of the channel configurations and
the SNR's. The first of these strategies is simple partial decode-and-forward.
Here, the source node uses a superposition codebook to broadcast independent
messages to the relays at appropriately chosen rates; each relay decodes its
intended message and then forwards it to the destination over the MAC channel.
A similar performance can be also achieved with compress-and-forward type
strategies (such as quantize-map-and-forward and noisy network coding) that
provide the 1.3 N-bit approximation for general Gaussian networks, but only if
the relays quantize their observed signals at a resolution inversely
proportional to the number of relay nodes N. This suggest that the
rule-of-thumb to quantize the received signals at the noise level in the
current literature can be highly suboptimal.
