A general framework for codes involving redundancy minimization

Abstract

A framework with two scalar parameters is introduced for various problems of finding a prefix code minimizing a coding penalty function. The framework encompasses problems previously proposed by Huffman, Campbell, Nath, and Drmota and Szpankowski, shedding light on the relationships among these problems. In particular, Nath's range of problems can be seen as bridging the minimum average redundancy problem of Huffman with the minimum maximum pointwise redundancy problem of Drmota and Szpankowski. Using this framework, two linear-time Huffman-like algorithms are devised for the minimum maximum pointwise redundancy problem, the only one in the framework not previously solved with a Huffman-like algorithm. Both algorithms provide solutions common to this problem and a subrange of Nath's problems, the second algorithm being distinguished by its ability to find the minimum variance solution among all solutions common to the minimum maximum pointwise redundancy and Nath problems. Simple redundancy bounds are also presented.

Optimal Prefix Codes for Infinite Alphabets With Nonlinear Costs

“…We show in Appendix I that lim inf θ↑1 R * (N * * θ , P θ ) = 1 − log 2 log 2 e and lim sup θ↑1 R * (N * * θ , P θ ) = 2 − log 2 e, and we characterize this oscillating behavior. This technique is extensible to other redundancy scenarios of the kind introduced in [23].…”

“…First note that this requires, as a necessary step, the ability to construct a minimum maximal pointwise redundancy code for finite alphabets. This can be done either with the method in [34] or any of those in [23], the simplest of which uses a variant of the tree-height problem [19], solved via a different extension of Huffman coding. Simply put, the weight combining rule, rather than w(j) + w(k) or a · (w(j) + w(k)), is…”

Infinite-Alphabet Prefix Codes Optimal for ß-Exponential Penalties

“…REDUNDANCY PENALTIES It is natural to ask whether the above results can be extended to other penalties. One penalty discussed in the literature is that of maximal pointwise redundancy [24], in which one seeks to find a code to minimize R*(N,P) A max [n(i)+log2p(i)] This can be shown to be a limit of the exponential case, as in [25], allowing us to analyze it using the same techniques as exponential Huffman coding. This limit can be shown by defining dth exponential redundancy as follows: Rd (N, P) A d 1g2 EiX P()2d(n(i)±og2p(i)) d°2Ei,X p(i)1l+d dn(i) Thus R* (N, P) = limd,0 Rd (N, P), and the above methods should apply in the limit.…”

Balancing decoding speed and memory usage for Huffman codes using quaternary tree

“…It shows that the performance is always declining when moving to the lower extension to higher extension. Szpankowski (2011) and Baer (2006) explained the minimum expected length of fixed-to-variable lossless compression without prefix constraint. Huffman principle, which is well known for fixed-to-variable code, is used in Kavousianos (2008) as a variable-to-variable code.…”