Tree Compression with Top Trees Revisited

We revisit tree compression with top trees (Bille et al, ICALP'13) and present several improvements to the compressor and its analysis. By significantly reducing the amount of information stored and guiding the compression step using a RePair-inspired heuristic, we obtain a fast compressor achieving good compression ratios, addressing an open problem posed by Bille et al. We show how, with relatively small overhead, the compressed file can be converted into an in-memory representation that supports basic navigation operations in worst-case logarithmic time without decompression. We also show a much improved worst-case bound on the size of the output of top-tree compression (answering an open question posed in a talk on this algorithm by Weimann in 2012).Comment: SEA 201

Arithmetic coding revisited

Over the last decade, arithmetic coding has emerged as an important compression tool. It is now the method of choice for adaptive coding on multisymbol alphabets because of its speed, low storage requirements, and effectiveness of compression. This article describes a new implementation of arithmetic coding that incorporates several improvements over a widely used earlier version by Witten, Neal, and Cleary, which has become a de facto standard. These improvements include fewer multiplicative operations, greatly extended range of alphabet sizes and symbol probabilities, and the use of low-precision arithmetic, permitting implementation by fast shift/add operations. We also describe a modular structure that separates the coding, modeling, and probability estimation components of a compression system. To motivate the improved coder, we consider the needs of a word-based text compression program. We report a range of experimental results using this and other models. Complete source code is available

Wavelet Trees Meet Suffix Trees

We present an improved wavelet tree construction algorithm and discuss its applications to a number of rank/select problems for integer keys and strings. Given a string of length n over an alphabet of size $\sigma\leq n$, our method builds the wavelet tree in $O(n \log \sigma/ \sqrt{\log{n}})$ time, improving upon the state-of-the-art algorithm by a factor of $\sqrt{\log n}$. As a consequence, given an array of n integers we can construct in $O(n \sqrt{\log n})$ time a data structure consisting of $O(n)$ machine words and capable of answering rank/select queries for the subranges of the array in $O(\log n / \log \log n)$ time. This is a $\log \log n$-factor improvement in query time compared to Chan and P\u{a}tra\c{s}cu and a $\sqrt{\log n}$-factor improvement in construction time compared to Brodal et al. Next, we switch to stringological context and propose a novel notion of wavelet suffix trees. For a string w of length n, this data structure occupies $O(n)$ words, takes $O(n \sqrt{\log n})$ time to construct, and simultaneously captures the combinatorial structure of substrings of w while enabling efficient top-down traversal and binary search. In particular, with a wavelet suffix tree we are able to answer in $O(\log |x|)$ time the following two natural analogues of rank/select queries for suffixes of substrings: for substrings x and y of w count the number of suffixes of x that are lexicographically smaller than y, and for a substring x of w and an integer k, find the k-th lexicographically smallest suffix of x. We further show that wavelet suffix trees allow to compute a run-length-encoded Burrows-Wheeler transform of a substring x of w in $O(s \log |x|)$ time, where s denotes the length of the resulting run-length encoding. This answers a question by Cormode and Muthukrishnan, who considered an analogous problem for Lempel-Ziv compression.Comment: 33 pages, 5 figures; preliminary version published at SODA 201