Hypertrellis: A Generalization of Trellis and Factor Graph

Abstract

Factor graphs have recently been introduced as an efficient graphical model for codes to study iterative decoding algorithms. However, it is well-known that a factor graph generalizes only the time axis of a trellis, but omits the state transition representation. In this paper, a new graphical model, called the hypertrellis, is proposed to overcome this insufficiency of factor graphs. A hypertrellis is in essence a weighted hypergraph generalization of a traditional trellis. Its time topology, which extends the time axis of a trellis, can take the form of any factor graph. A key to this extension is the interpretation of a factor graph as a factor hypergraph. This is facilitated by introducing a “starfish” drawing representation for hypergraphs, which enhances the applicability of hypergraph models by enabling simpler drawing and easier visualization, relative to the traditional representation. The maximum likelihood decoding (MLD) problem is then formulated as a shortest hyperpath search on a hypertrellis. For hypertrellises with an acyclic time topology, a hyperpath-oriented MLD algorithm, called the one-way algorithm, is introduced. The one-way algorithm, as a hypertrellis generalization of the celebrated Viterbi algorithm, provides insights into efficient management of the surviving hyperpath history and various practical hyperpath-oriented simplifications. It is shown that as a MLD algorithm, the one-way algorithm has a lower minimal decoding delay. The computational complexity and the amount of storage needed are also bett er than with the well-known min-sum algorithm. Some connections between the hypertrellis and another recently proposed bipartite graph model, called the trellis formation, are also discussed.