Reserved-length prefix coding

Abstract

Abstract-Huffman coding finds an optimal prefix code for a given probability mass function. Consider situations in which one wishes to find an optimal code with the restriction that all codewords have lengths that lie in a user-specified set of lengths (or, equivalently, no codewords have lengths that lie in a complementary set). This paper introduces a polynomial-time dynamic programming algorithm that finds optimal codes for this reserved-length prefix coding problem. This has applications to quickly encoding and decoding lossless codes. In addition, one modification of the approach solves any quasiarithmetic prefix coding problem, while another finds optimal codes restricted to the set of codes with g codeword lengths for user-specified g (e.g., g = 2).

A Generic Top-Down Dynamic-Programming Approach to Prefix-Free Coding

“…Baer [4] introduces these problems in the context of fast decoding and used a top-down DP approach to solve the first one in O(|Λ|n 3 ) time and the second one in O(n 3 g 3 log g n) time . We will reduce these two cases, respectively, to O(|Λ|n 2 ) and O(n 2 g log n) time.…”

“…Since variations of this formulation have been extensively used before for various coding problems, e.g. [15,12,10,4], we only sketch the method but do not rigorously prove its correctness.…”

“…Since there are at most g levels, our generic GMR algorithm solves this problem in O(gn 2 ) time, improving upon the O(gn 3 ) algorithm of [4] .…”

“…Some examples of such prefix-free coding problems are Length-limited coding e.g, [16,17,3,18], Unequalcost coding e.g., [6,13,7,12] Mixed-radix coding [10], Reserved-length coding [4], and One-ended coding [5,8,9], The major observation is that all of the best algorithms known for these problems use some form of dynamic programming (DP) to build an optimal (mincost) coding tree that corresponds to an optimal code. These DPs primarily differ in whether they build the tree from the bottom-up or the top-down.…”

“…Some examples of such prefix-free coding problems are Length-limited coding e.g, [17,18,3,19], Unequalcost coding e.g., [7,15,8,13] Mixed-radix coding [11], Reserved-length coding [4], and One-ended coding [6,9,10],…”