Efficient alphabet partitioning algorithms for low-complexity entropy coding

We analyze the technique for reducing the complexity of entropy coding consisting of the a priori grouping of the source alphabet symbols, and in dividing the coding process in two stages: first coding the number of the symbol´s group with a more complex method, followed by coding the symbol´s rank inside its group using a less complex method, or simply using its binary representation. Because this method proved to be quite effective it is widely used in practice, and is an important part in standards like MPEG and JPEG. However, a theory to fully exploit its effectiveness had not been sufficiently developed. In this work, we study methods for optimizing the alphabet decomposition, and prove that a necessary optimality condition eliminates most of the possible solutions, and guarantees that dynamic programming solutions are optimal. In addition, we show that the data used for optimization have useful mathematical properties, which greatly reduce the complexity of finding optimal partitions. Finally, we extend the analysis, and propose efficient algorithms, for finding min-max optimal partitions for multiple data sources. Numerical results show the difference in redundancy for single and multiple sources.