Lower Bounds on the Redundancy of Huffman Codes with Known and Unknown Probabilities

In this paper, we provide a method to obtain tight lower bounds on the minimum redundancy achievable by a Huffman code when the probability distribution underlying an alphabet is only partially known. In particular, we address the case where the occurrence probabilities are unknown for some of the symbols in an alphabet. Bounds can be obtained for alphabets of a given size, for alphabets of up to a given size, and alphabets of arbitrary size. The method operates on a computer algebra system, yielding closed-form numbers for all results. Finally, we show the potential of the proposed method to shed some light on the structure of the minimum redundancy achievable by the Huffman code.