Restricted Hamming-Huffman Trees

Abstract

We study a special case of HammingHuffman trees, in which both data compression and data error detection are tackled on the same structure. Given a hypercube Qn of dimension n, we are interested in some aspects of its vertex neighborhoods. For a subset L of vertices of Qn, the neighborhood of L is defined as the union of the neighborhoods of the vertices of L. The minimum neighborhood problem is that of determining the minimum neighborhood cardinality over all those sets L. This is a well-known problem that has already been solved. Our interest lies in determining optimal HammingHuffman trees, a problem that remains open and which is related to minimum neighborhoods in Qn. In this work, we consider a restricted version of HammingHuffman trees, called [k]-HHT s, which admit symbol leaves in at most k different levels. We present an algorithm to build optimal [2]-HHT s. For uniform frequencies, we prove that an optimal HHT is always a [5]-HHT and that there exists an optimal HHT which is a [4]-HHT. Also, considering experimental results, we conjecture that there exists an optimal tree which is a [3]-HHT.