Universal Lossless Data Compression Via Binary Decision Diagrams

TL;DR

This paper introduces a lossless data compression method that encodes binary strings using their unique ROBDD representations, providing bounds on redundancy relative to source complexity.

Contribution

The paper presents a novel compression algorithm based on ROBDDs for binary strings of power-of-two length, with theoretical bounds on redundancy for any binary source.

Findings

Maximal redundancy per sample is bounded by (4 log2 s + 16 + o(1))/log2 n.

ROBDD representation size is on the order of (2+o(1)) 2^k / k vertices.

The method achieves lossless compression with provable redundancy bounds.

Abstract

A binary string of length $2^k$ induces the Boolean function of $k$ variables whose Shannon expansion is the given binary string. This Boolean function then is representable via a unique reduced ordered binary decision diagram (ROBDD). The given binary string is fully recoverable from this ROBDD. We exhibit a lossless data compression algorithm in which a binary string of length a power of two is compressed via compression of the ROBDD associated to it as described above. We show that when binary strings of length $n$ a power of two are compressed via this algorithm, the maximal pointwise redundancy/sample with respect to any s-state binary information source has the upper bound $(4\log_2s+16+o(1))/\log_2n $. To establish this result, we exploit a result of Liaw and Lin stating that the ROBDD representation of a Boolean function of $k$ variables contains a number of vertices on the order of $(2+o(1))2^{k}/k$.