A combinatorial approach to Golomb forests

Abstract

Optimal binary prefix-free codes for infinite sources with geometrically distributed frequencies, e.g., P={p i(1−p)} i=0 ∞, 0<p<1 , were first (implicitly) suggested by Golomb over 30 years ago in the context of run-length encodings. Ten years later Gallager and Van Voorhis exhibited such optimal codes for all values of p. These codes were derived by using the Huffman encoding algorithm to build optimal codes for finite sources and then showing that the finite codes converge in a very specific sense to the infinite one. In this note, we present a new combinatorial approach to solve the same problem, one that does not use the Huffman algorithm, but instead treats a coding tree as an infinite sequence of integers and derives properties of the sequence. One consequence of this new approach is a complete characterization of all of the optimal codes; in particular, it shows that for all p,0<p<1, except for an easily describable countable set, there is a unique optimal code, but for each p in this countable set there are an uncountable number of optimal codes. Another consequence is a derivation of infinite codes for geometric sources when the encoding alphabet is no longer restricted to be the binary one. A final consequence is the extension of the results to optimal forests instead of being restricted to optimal trees.