Monotonicity-Based Fast Algorithms for MAP Estimation of Markov Sequences Over Noisy Channels

In this correspondence, we study algorithmic approach to solving the problem of maximum a posteriori (MAP) estimation of Markov sequences transmitted over noisy channels, which is also known as the MAP decoding problem. For the class of memoryless binary channels that produce independent substitution and erasure errors, the MAP sequence estimation problem can be formulated and solved as one of the longest path in a weighted directed acyclic graph. But for algorithm efficiency, we transform the graph problem to one of matrix search. If the underlying matrix is totally monotone, then the complexity of MAP sequence estimation can be greatly reduced. We give a sufficient condition for the matrix induced by MAP sequence estimation to be totally monotone, which is indeed the case if the input sequence is Gaussian Markov. Under this condition, the complexity of MAP decoding can be reduced from $O(N^{2}M)$ to $O(NM)$, where $N$ is the size of source alphabet and $M$ is the length of input sequence. Furthermore, for Markov sequences of fixed-length code we propose a block parsing strategy to reduce the complexity of MAP sequence estimation to $O(M + N^{2} M / \log M)$ or to $O(M + N M / \log M)$, depending on if the total monotonicity holds. Another significance of this correspondence lies in the applicability of the presented algorithmic approach, which has been thoroughly studied in computer science literature, to many other discrete optimization problems encountered in both source and channel coding, ranging from optimal multiresolution and multiple-description quantizer design, to context quantization for minimum conditional entropy, and to optimal packetization with uneven error protection.