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 <inline-formula> <tex-math notation="LaTeX">\mathbf {x} = (x_{1}, x_{2}, \ldots, x_{n}) </tex-math></inline-formula> and <inline-formula>...

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Bibliographic Details
Published in:IEEE transactions on information theory Vol. 68; no. 1; pp. 93 - 104
Main Authors: He, Xuan, Cai, Kui, Zhou, Liang
Format: Journal Article
Language:English
Published: New York IEEE 01.01.2022
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Algorithms
Binary structure
Binary trees
Complexity
Complexity theory
Decoding
Directed graphs
Electronic mail
latency
low-density parity-check (LDPC) code
Message passing
message passing algorithm
Nodes
Parity check codes
ISSN:0018-9448, 1557-9654
Online Access:Get full text
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Summary:Consider the computations at a node in a message passing algorithm. Assume that the node has incoming and outgoing messages <inline-formula> <tex-math notation="LaTeX">\mathbf {x} = (x_{1}, x_{2}, \ldots, x_{n}) </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">\mathbf {y} = (y_{1}, y_{2}, \ldots, y_{n}) </tex-math></inline-formula>, respectively. In this paper, we investigate a class of structures that can be adopted by the node for computing <inline-formula> <tex-math notation="LaTeX">\mathbf {y} </tex-math></inline-formula> from <inline-formula> <tex-math notation="LaTeX">\mathbf {x} </tex-math></inline-formula>, where each <inline-formula> <tex-math notation="LaTeX">y_{j}, j = 1, 2, \ldots, n </tex-math></inline-formula> is computed via a binary tree with leaves <inline-formula> <tex-math notation="LaTeX">\mathbf {x} </tex-math></inline-formula> excluding <inline-formula> <tex-math notation="LaTeX">x_{j} </tex-math></inline-formula>. We make three main contributions regarding this class of structures. First, we prove that the minimum complexity of such a structure is <inline-formula> <tex-math notation="LaTeX">3n - 6 </tex-math></inline-formula>, and if a structure has such complexity, its minimum latency is <inline-formula> <tex-math notation="LaTeX">\delta + \lceil \log (n-2^{\delta }) \rceil </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">\delta = \lfloor \log (n/2) \rfloor </tex-math></inline-formula>, where the logarithm always takes base two. Second, we prove that the minimum latency of such a structure is <inline-formula> <tex-math notation="LaTeX">\lceil \log (n-1) \rceil </tex-math></inline-formula>, and if a structure has such latency, its minimum complexity is <inline-formula> <tex-math notation="LaTeX">n \log (n-1) </tex-math></inline-formula> when <inline-formula> <tex-math notation="LaTeX">n-1 </tex-math></inline-formula> is a power of two. Third, given <inline-formula> <tex-math notation="LaTeX">(n, \tau) </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">\tau \geq \lceil \log (n-1) \rceil </tex-math></inline-formula>, we propose a construction for a structure which we conjecture to have the minimum complexity among structures with latencies at most <inline-formula> <tex-math notation="LaTeX">\tau </tex-math></inline-formula>. Our construction method runs in <inline-formula> <tex-math notation="LaTeX">O(n^{3} \log ^{2}(n)) </tex-math></inline-formula> time, and the obtained structure has complexity at most (generally much smaller than) <inline-formula> <tex-math notation="LaTeX">n \lceil \log (n) \rceil - 2 </tex-math></inline-formula>.
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2021.3119952