D-ary Bounded-Length Huffman Coding

Abstract

Abstract-Efficient optimal prefix coding has long been accomplished via the Huffman algorithm. However, there is still room for improvement and exploration regarding variants of the Huffman problem. Length-limited Huffman coding, useful for many practical applications, is one such variant, in which codes are restricted to the set of codes in which none of the n codewords is longer than a given length, lmax. Binary lengthlimited coding can be done in O(nlmax) time and O(n) space via the widely used Package-Merge algorithm. In this paper the Package-Merge approach is generalized without increasing complexity in order to introduce a minimum codeword length, lmin, to allow for objective functions other than the minimization of expected codeword length, and to be applicable to both binary and nonbinary codes; nonbinary codes were previously addressed using a slower dynamic programming approach. These extensions have various applications -including faster decompressionand can be used to solve the problem of finding an optimal code with limited fringe, that is, finding the best code among codes with a maximum difference between the longest and shortest codewords. The previously proposed method for solving this problem was nonpolynomial time, whereas solving this using the novel algorithm requires only O(n(lmax − lmin)2 ) time and O(n) space.

On compressing permutations and adaptive sorting

“…The multiary case is far less understood than the binary case. Recently, an algorithm to find the optimal length-restricted t-ary code has been presented whose running time is linear once the lengths are sorted [Bae07]. To analyze the increase in redundancy, consider the sub-optimal method that simply takes any node v of depth more than ℓ = 4 lg nRuns/ lg t and balances its subtree (so that height 5 lg nRuns/ lg t is guaranteed).…”

The Number of Huffman Codes, Compact Trees, and Sums of Unit Fractions

“…for b = a − 31a 2 /60 > 0. By (4.2a) and (4.2b), we have 1 |z| [3] So we now assume that D(1/z) = 0 for some z with r 3 ≤ |z| < 1 + log 2/t. Repeating the above steps with 1/t 3 replaced by 0 gives the slightly better bound z = ρ 2 with ρ 2 as in (2.3).…”

Source Coding for Quasiarithmetic Penalties

“…However, consider nonbinary output alphabets, that is, D > 2. As in Huffman coding for such alphabets, we must add "dummy" with proof of correctness, in [21].…”

“…Although we briefly sketch how to adapt this technique to general output alphabet coding at the end of Subsection IV-E, an approach fully explained in [21], until then we concentrate on the binary case (D = 2).…”

“…With this preprocessing step, finding an optimal code should proceed similarly to the binary case, with adjustments made for both the Package-Merge algorithm and the overall coding algorithm due to the formulation of the Kraft inequality and maximum length. A complete algorithm is available, with proof of correctness, in [21].…”