Decision Procedure for the Existence of Two-Channel Prefix-Free Codes

Abstract

The Kraft inequality gives a necessary and sufficient condition for the existence of a single channel prefix-free code. However, the multichannel Kraft inequality does not imply the existence of a multichannel prefix-free code in general. It is natural to ask whatever there exists an efficient decision procedure for the existence of multichannel prefixfree codes. In this paper, we tackle the two-channel case of the above problem by relating it to a constrained rectangle packing problem. Although a general rectangle packing problem is NP-complete, the extra imposed constraints allow us to propose an algorithm which can solve the problem efficiently.

On Multi-Channel Huffman Codes for Asymmetric-Alphabet Channels

“…Specifically, we consider the network in Fig. 1 and compare the expected codeword (description) length 2 of an optimal (2, 3)-ary tree-decodable code 3 , the binary Huffman code, and the ternary Huffman code for the information sources with probability masses {1/6, 1/6, 1/3, 1/3} and {1/6, 1/6, 1/6, 1/2} respectively in the following examples. For convenience, we use the information unit nat (base e, the Euler's number) throughout this paper.…”

“…The definition of a multi-channel source code was proposed in [2] for symmetric-alphabet channels and extended in [3] for asymmetric-alphabet channels. Consider an n-channel system where the ith channel uses an alphabet Z i of size q i , i ∈ [n].…”

“…By convention, a prefix-free code is also called a prefix code. Any codeword of a uniquely decodable multi-channel source code can be decoded without referring to the symbols of any future codewords if and only if the code is a multi-channel prefix code [2], [3].…”

“…The multi-channel analog to Kraft inequality and entropy bound were proposed in [2] for symmetric-alphabet channels. They were extended for asymmetric-alphabet channels in the conference paper [3] but their proofs were separated in the preprint [5]. For the sake of completeness, we reproduce the proofs in this paper.…”

“…Theorem 1 (Kraft Inequality [3]). If Q is uniquely decodable, then the length tuples of its codewords satisfy…”