Asymmetric Encoding-Decoding Schemes for Lossless Data Compression

TL;DR

The paper introduces the asymmetric encoding-decoding scheme (AEDS), a broad generalization of the tANS that uses backward encoding and forward decoding but without the strict arithmetic constraints of conventional coding. It develops concrete constructions from code trees (Type-I and Type-II AEDS) and shows that, for i.i.d. sources, AEDS can outperform Huffman when the right-subtree probability weight exceeds specific thresholds, with explicit reductions in average code length. It extends the framework to state-divided AEDS (sAEDS), derives practical upper bounds on the average length for different divisibility scenarios, and proves that the optimal sAEDS and tANS converge to the source entropy at a rate of $O(1/N)$ as the number of states grows. The results provide both design strategies for efficient, fast, table-based coders and theoretical guarantees on compression performance, including in the uniform and binary-source cases, demonstrating potential practical gains over Huffman coding with modest state counts.

Abstract

This paper proposes a new lossless data compression coding scheme named an asymmetric encoding-decoding scheme (AEDS), which can be considered as a generalization of tANS (tabled variant of asymmetric numeral systems). In the AEDS, a data sequence $\bm{s}=s_1s_2\cdots s_n$ is encoded in backward order $s_t, t=n, \cdots, 2,1$, while $\bm{s}$ is decoded in forward order $s_t, t=1, 2, \cdots, n$ in the same way as the tANS. But, the code class of the AEDS is much broader than that of the tANS. We show for i.i.d.~sources that an AEDS with 2 states (resp.~5 states) can attain a shorter average code length than the Huffman code if a child of the root in the Huffman code tree has a probability weight larger than 0.61803 (resp.~0.56984). Furthermore, we derive several upper bounds on the average code length of the AEDS, which also hold for the tANS, and we show that the average code length of the optimal AEDS and tANS with $N$ states converges to the source entropy with speed $O(1/N)$ as $N$ increases.

Figures (19)

• Figure 1: An example of an AEDS for $\mathcal{S}=\{a, b, c\}$, $\mathcal{X}=\{\alpha_1, \alpha_2, \cdots, \alpha_5\}$, and $N=5$.
• Figure 2: A code tree $T$. Any prefix-free code tree can be used to construct an AEDS. But, a Huffman code tree is used as $T$ in Corollaries 1 and 2.
• Figure 3: Type-I AEDS with $N$ states based on a code tree $T$, where $\mathcal{B}_{\alpha_1}^{(D)}$ is given by Fig. \ref{['fig-I2']} and $\mathcal{B}_{\alpha_j}^{(D)}=T_R$ for $2\leq j \leq N$. The right side of the figure indicates the codeword composition in encoding.
• Figure 4: The code tree of $\mathcal{B}_{\alpha_1}^{(D)}$ used in Type-I AEDS.
• Figure 5: The reduction $\delta_N^{(\mathrm{I})}(P_R)$ attained by Type-I AEDS in the average code length.

Theorems & Definitions (24)

• Theorem 1
• Theorem 2
• Corollary 1
• proof
• Corollary 2
• proof
• Remark 1
• Remark 2
• Remark 3
• Remark 4