Source Coding with Free Bits and the Multi-Way Number Partitioning Problem

TL;DR

This paper presents a novel source coding scheme involving free bits over two channels, relates it to the multi-way number partitioning problem, and provides efficient algorithms for optimal and near-optimal solutions.

Contribution

It introduces a new source coding variant with free bits, connects it to a generalized number partitioning problem, and develops efficient Huffman-based algorithms for optimal solutions.

Findings

Optimal solution achieved with an adapted Huffman algorithm.

Nearly optimal solution for a new objective function.

Runtime of the algorithm is O(n log n).

Abstract

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.