A Class of Optimal Structures for Node Computations in Message Passing Algorithms
Xuan He, Kui Cai, Liang Zhou · IEEE Transactions on Information Theory · 2021
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$.