Two Classes of Optimal Multi-Input Structures for Node Computations in Message Passing Algorithms

TL;DR

This work tackles efficient node updates in message-passing algorithms by introducing two structure classes tuned to different priorities: star-tree-based structures for complexity-centric design and isomorphic-DRT-based structures for latency-centric design. It provides a rigorous optimization framework, deriving feasibility conditions $2+\sum_{i=1}^{m-1} i q_i= n$ and $\prod_{i=1}^{m-1} (i+1)^{w_i}=n-1$, and developing DP algorithms that achieve near-optimal or optimal performance under these constraints, with complexities $O(mn)$ and $O(m\prod_{i=1}^{m-1}(q_i+1))$ respectively. The results establish that star-tree-based structures can achieve the lowest complexity (and, in some cases, the lowest latency among those), while isomorphic-DRT-based structures can achieve the absolute minimum latency, with an explicit method to realize the latency-optimal structures at the lowest complexity. This work extends binary-input insights to multi-input scenarios and offers practical avenues for hardware-efficient, low-latency implementations of message-passing decoders and related algorithms.

Abstract

In this paper, we delve into the computations performed at a node within a message-passing algorithm. We investigate low complexity/latency multi-input structures that can be adopted by the node for computing outgoing messages y = (y1, y2, . . . , yn) from incoming messages x = (x1, x2, . . . , xn), where each yj , j = 1, 2, . . . , n is computed via a multi-way tree with leaves x excluding xj . Specifically, we propose two classes of structures for different scenarios. For the scenario where complexity has a higher priority than latency, the star-tree-based structures are proposed. The complexity-optimal ones (as well as their lowest latency) of such structures are obtained, which have the near-lowest (and sometimes the lowest) complexity among all structures. For the scenario where latency has a higher priority than complexity, the isomorphic-directed-rooted-tree-based structures are proposed. The latency-optimal ones (as well as their lowest complexity) of such structures are obtained, which are proved to have the lowest latency among all structures.

Figures (9)

• Figure 1: Message passing on a specific node.
• Figure 2: DRTs rooted at $\mathbf{y}$ with $n=7$.
• Figure 3: A structure in Ueng2017 for computation of $\mathbf{y}$ with $n=7$.
• Figure 4: A latency-optimal isomorphic-DRT-based structure for computation of $\mathbf{y}$ with $n=7$.
• Figure 5: The only star tree $T \in \mathcal{T}_3$.

Theorems & Definitions (20)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5: star trees
• Lemma 1
• Lemma 2
• Lemma 3
• Theorem 1
• Lemma 4