Minimum codeword length and redundancy of Huffman codes

Abstract

A tight upper bound on the redundancy r of Huffman codes, in terms of the minimum codeword length l, l≥1, is provided. The bound is a strictly decreasing function of l. For large l it yields r≤l−log(2^l+1−1)+1+β+O(2^−2l), where β≈0.0860.

By using this result we improve Gallager's bound on the redundancy when only the most likely source probability p ₁ is known.

This work was partially supported by the National Council of Research (C.N.R.) under grant 90.01552.CT12.