A Class of Optimal Structures for Node Computations in Message Passing Algorithms

Consider the computations at a node in a message passing algorithm. Assume that the node has incoming and outgoing messages
$\mathbf {x} = (x_{1}, x_{2}, \ldots, x_{n})$
and
$\mathbf {y} = (y_{1}, y_{2}, \ldots, y_{n})$
, respectively. In this paper, we investigate a class of structures that can be adopted by the node for computing
$\mathbf {y}$
from
$\mathbf {x}$
, where each
$y_{j}, j = 1, 2, \ldots, n$
is computed via a binary tree with leaves
$\mathbf {x}$
excluding
$x_{j}$
. We make three main contributions regarding this class of structures. First, we prove that the minimum complexity of such a structure is
$3n - 6$
, and if a structure has such complexity, its minimum latency is
$\delta + \lceil \log (n-2^{\delta }) \rceil $
with
$\delta = \lfloor \log (n/2) \rfloor $
, where the logarithm always takes base two. Second, we prove that the minimum latency of such a structure is
$\lceil \log (n-1) \rceil $
, and if a structure has such latency, its minimum complexity is
$n \log (n-1)$
when
$n-1$
is a power of two. Third, given
$(n, \tau)$
with
$\tau \geq \lceil \log (n-1) \rceil $
, we propose a construction for a structure which we conjecture to have the minimum complexity among structures with latencies at most
$\tau $
. Our construction method runs in
$O(n^{3} \log ^{2}(n))$
time, and the obtained structure has complexity at most (generally much smaller than)
$n \lceil \log (n) \rceil - 2$
.