Compressed Dynamic Tries with Applications to LZ-Compression in Sublinear Time and Space

Abstract

The dynamic trie is a fundamental data structure which finds applications in many areas. This paper proposes a compressed version of the dynamic trie data structure. Our data-structure is not only space efficient, it also allows pattern searching in o(|P|) time and leaf insertion/deletion in o(logn) time, where |P| is the length of the pattern and n is the size of the trie. To demonstrate the usefulness of the new data structure, we apply it to the LZ-compression problem. For a string S of length s over an alphabet $\cal A$ of size σ, the previously best known algorithms for computing the Ziv-Lempel encoding (lz78) of S either run in: (1) O(s) time and O(s logs) bits working space; or (2) O(s σ) time and O(s H _k  + s logσ/ log _σ s) bits working space, where H _k is the k-order entropy of the text. No previous algorithm runs in sublinear time. Our new data structure implies a LZ-compression algorithm which runs in sublinear time and uses optimal working space. More precisely, the LZ-compression algorithm uses O(s (logσ + loglog _σ s) / log _σ s) bits working space and runs in $O(s (\log \log s)^2 / (\log_{\sigma} s \log \log \log s))$ worst-case time, which is sublinear when $\sigma = 2^{o(\log s \frac{\log \log \log s}{(\log \log s)^2})}$.