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) $n \lceil \log(n) \rceil - 2$.

翻译：考虑在消息传递算法的节点计算 。 假设节点的收发和发送信息 $\ mathbf{x} = (x_ 1, x_ 2,\ ldot, x_n) $ 和 $\ mathbf{y} = (y_ 1, y2, eldots, y_n) = (y_ 2, e) 。 在本文件中, 我们调查节点可以采用的一组结构, 计算 $ mathbf{y} $( mathbf{x} $ $, 其中每个 $_ j, j= 1, 2,\ ldolots, 美元 美元结构计算 $xxx$xx_ 美元。 我们为这一结构的最小复杂性为 3n - 6 美元, 如果一个结构如此复杂, 其最小值是 $- lcrequet $ (delta+ lcreal) (n-2 $) ral\ ral\ roum a laum a.