Dynamic programming and the graphical representation of error-correcting codes

Abstract

Abstract-Graphical representations of codes facilitate the design of computationally efficient decoding algorithms. This is an example of a general connection between dependency graphs, as arise in the representations of Markov random fields, and the dynamic programming principle. We concentrate on two computational tasks: finding the maximum-likelihood codeword and finding its posterior probability, given a signal received through a noisy channel. These two computations lend themselves to a particularly elegant version of dynamic programming, whereby the decoding complexity is particularly transparent. We explore some codes and some graphical representations designed specifically to facilitate computation. We further explore a coarse-to-fine version of dynamic programming that can produce an exact maximum-likelihood decoding many orders of magnitude faster than ordinary dynamic programming.

On the resolution of local ambiguity in visual recognition: Modeling by tree-structured Markov random fields

“…Successful examples of object recognition include mine detection by Raphael [26] and recognition of mathematical formulas by Chou [27]. There is an application to error-correcting codes in which the information is embedded in each three nodes in a tree-structured MRF [28].…”

Non-parametric Decoding on Discrete Time Series and Its Applications in Bioinformatics

“…For example, the most widely used decoding technique is the Viterbi algorithm [29]. Its computation of the most likely configuration can be achieved within O(n|S| 2 ) operations [12]. Now several of its variant algorithms are also popular for different purposes, such as the posterior-Viterbi algorithm.…”

“…This corresponds to a very simple linear graphic structure sequentially linking S(1) through S(n). See Geman and Kochanek [12] for an example of a HMM represented by a more complex graph.…”

“…It is used to compute a sequence of exons and introns given a genome sequence in computational genetics [5]. And as mentioned, it is used in computing a sequence of messages given an output signal sequence in information theory [12].…”

“…Decoding is most often referred to in information theory as an algorithm for recovering a sequence of code words or messages, given a sequence of output signals in communication through a noise channel [12]. It is a task corresponding to attempts to understand non-autonomous dynamics in mathematics and physics [21].…”