Twenty (or so) Questions: D-ary Bounded-Length Huffman Coding

The game of Twenty Questions has long been used to illustrate binary source coding. Recently, a physical device has been developed that mimics the process of playing Twenty Questions, with the device supplying the questions and the user providing the answers. However, this game differs from Twenty Questions in two ways: Answers need not be only “yes” and “no,” and the device continues to ask questions beyond the traditional twenty; typically, at least 20 and at most 25 questions are asked. The nonbinary variation on source coding is one that is well known and understood, but not with such bounds on length. An O(n(lmax− lmin))-time O(n)-space PackageMerge-based algorithm is presented here for D-ary (binary or nonbinary) source coding with all n codeword lengths (numbers of questions) bounded to be within the interval [lmin, lmax]. This algorithm minimizes average codeword length or, more generally, any other quasiarithmetic convex coding penalty. In the case of minimizing average codeword length, both time and space complexity can be improved via an alternative graph-based reduction. This has, as a special case, a method for nonbinary length-limited Huffman coding, which was previously solved via dynamic programming with O(n2lmax logD) time and O(n logD) space. These algorithms can also be used to efficiently find a code that is optimal given a limit on fringe, the difference between the lengths of longest and shortest codewords, a problem previously without a polynomial-time solution.