Local redundancy and progressive bounds on the redundancy of a Huffman code

The concept of local redundancy is introduced, and it is shown that it is a simple but powerful tool for understanding the redundancy of a prefix code. Based on this concept, the redundancy of a prefix code when the structure of the code is partially known is proven. It is shown how this theorem can be used to obtain well-known lower bounds on R_Huff, the redundancy of a Huffman code. The theorem is used to obtain a sequence of new lower bounds on R_Huff based on the Huffman procedure. A similar sequence of upper bounds on R_Huff is obtained. New lower and upper bounds on R_Huff when the two smallest probabilities of the source distribution are known are given as corollaries of these results.