Non-Binary Robust Universal Variable Length Codes

The famous Fibonacci series defined by F ₀ = 0, F ₁ = 1 and F _i =F _i-1 + F _i-2 , for i ≥ 2, has contributed to several applications in Data Compression. Quite a few encodings have been suggested that are based on the properties of the Fibonacci sequence, and can be used as alternatives to fixed length codes. Interest in the Fibonacci sequence has shifted also to other adjacent areas, such as compressed matching[1], in which a pattern P is to be located in a text T, which is assumed to be given in some compressed form. The advantage of using Fibonacci codes in this context is the 11 separator, that acts as a border between adjacent codewords. Compressed matching is not limited to text files only, and in [3] the Fibonacci code was adapted to tools in image compression. Codes of order d are related to d-ary trees and there are several motivations for using such trees with d > 2. A d-ary tree is of height [log _d |Σ|], which may improve the processing time for larger d, e.g., in higher order Wavelet trees, where instead of storing binary bitmaps in every internal node, one rather stores sequences over the alphabet {1,...,d}. Ferragina et al.[2] show how to handle rank and select of such sequences, that improve both time and space complexities. A Fibonacci Wavelet tree, in which pruning was applied for additional savings, was defined in [4]. This data structure has then been generalized to higher-order of another kind, namely by the use of higher order Fibonacci Codes in which each element of the underlying sequence is the sum of the k preceding ones, for k ≥ 2. The corresponding binary code has the property that there is no occurrence of a string of m consecutive 1s. We extend the binary Fibonacci code to d-ary codes, with d ≥ 2. This is motivated by future technological developments in which the basic unit of storage will not be just a 2-valued bit, but possibly an element that is able to distinguish between d different values. The proposed codes are prefix-free, complete and more robust than Huffman codes. Experimental results illustrate that the compression efficiency of non-binary Fibonacci codes are very close to the savings achieved by the corresponding non-binary Huffman coding of the same order.