A Gallager-Type Redundancy Bound for Binary Shannon-Fano Coding

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Researchers have established a new redundancy bound for binary Shannon-Fano coding that depends on the largest source probability $p_1$, showing an explicit seven-piece envelope with a maximum redundancy $R<0.5651$ for $p_1<\frac{1}{2}$, marking the first $p_1$-dependent bound for Fano codes and utilizing a more sophisticated method than traditional Huffman coding approaches.

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Excerpt

arXiv:2607.04192v1 Announce Type: new Abstract: Kraj\v{c}i, Liu, Mike\v{s}, and Moser proved in 2015 that the redundancy of binary Shannon-Fano coding is always below one bit. We sharpen this to a bound depending on the largest source probability $p_1$: an explicit seven-piece envelope $R<\tfrac52-\tfrac56\log_2 5=0.5651$ for $p_1<\tfrac12$. It is the first $p_1$-dependent redundancy bound for Fano codes. The method is more sophisticated than the approach typical for Huffman codes: Fano trees are built top-down by contiguous balanced splits and lack the sibling property. From the $R<1$ theorem the rest follows from the Fano recursion, through a min-corrected affine potential and a no-burial lemma. Every scalar inequality in the proof reduces to a comparison of integer powers.