Consider the computations at a node in the message passing algorithms. 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 have 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 $\delta = \lfloor \log(n/2) \rfloor$. 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 likely has 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) $n \lceil \log(n) \rceil - 2$.

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一类消息传递算法中节点计算的最优结构

考虑消息传递算法中节点处的计算。假设节点分别有传入和传出消息 $\mathbf{x} = (x_1, x_2, \ldots, x_n)$ 和 $\mathbf{y} = (y_1, y_2, \ldots, y_n)$。在本文中，我们研究了节点可以采用的一类结构，用于从 $\mathbf{x}$ 计算 $\mathbf{y}$，其中每个 $y_j, j = 1, 2, \ldots, n $ 是通过带有叶子 $\mathbf{x}$ 的二叉树计算的，不包括 $x_j$。我们对此类结构有三个主要贡献。首先，我们证明这样一个结构的最小复杂度是$3n - 6$，如果一个结构有这样的复杂度，它的最小延迟是$\delta + \lceil \log(n-2^{\delta}) \ rceil$ 与 $\delta = \lfloor \log(n/2) \rfloor$。其次，我们证明这种结构的最小延迟是 $\lceil \log(n-1) \rceil$，如果一个结构有这样的延迟，当 $n-1$ 是 2 的幂时，它的最小复杂度是 $n \log(n-1)$。第三，给定 $(n, \tau)$ 和 $\tau \geq \lceil \log(n-1) \rceil$，我们提出了一个结构的构造，它可能在延迟最多 $ 的结构中具有最小的复杂性\tau$。我们的构造方法在$O(n^3 \log^2(n))$时间内运行，得到的结构最多复杂度（一般远小于）$n \lceil \log(n) \rceil - 2$ .