Variable-to-Variable Huffman Coding: Optimal and Greedy Approaches

In this paper, we address the problem of m-gram entropy variable-to-variable coding, extending the classical Huffman algorithm to the case of coding m-element (i.e., m-grams) sequences of symbols taken from the stream of input data for m>1. We propose a procedure to enable the determination of the frequencies of the occurrence of m-grams in the input data; we formulate the optimal coding algorithm and estimate its computational complexity as O(mn2), where n is the size of the input data. Since such complexity is high in terms of practical applications, we also propose an approximate approach with linear complexity, which is based on a greedy heuristic used in solving backpack problems. In order to verify the practical effectiveness of the proposed approximate approach, experiments involving different sets of input data were conducted. The experimental study shows that the results obtained with the approximate approach were, first, close to the optimal results and, second, better than the results obtained using the popular DEFLATE and PPM algorithms in the case of data that can be characterized by highly invariable and easy to estimate statistics.