We introduce a new variant of variable-length source coding for sending a source over two parallel channels, one of which is costly and the other free. We give a complete solution to this problem. Next, we relate the problem to the number partitioning problem, which is the task of dividing a given list of numbers into a pre-specified number of subsets such that the sum of the numbers in each subset is as nearly equal as possible. We introduce two new objective functions for this problem and show that an adapted version of the Huffman coding algorithm (with a runtime of $\mathcal{O}(n \log n)$ for input size $n$) produces the optimal solution for one objective function, and a nearly optimal solution for the other objective function.

翻译：本文提出了一种新的变长源编码方案，用于通过两条并行信道传输信源，其中一条信道成本高昂，另一条则免费。我们给出了该问题的完整解决方案。随后，我们将该问题与数字划分问题联系起来，后者旨在将给定数字列表划分为预定数量的子集，使得各子集内数字之和尽可能接近。针对该问题，我们引入了两个新的目标函数，并证明采用改进的Huffman编码算法（输入规模为$n$时时间复杂度为$\mathcal{O}(n \log n)$）能为其中一个目标函数生成最优解，对另一个目标函数则能产生近似最优解。