We show that computing the smallest run-length straight line program (RLSLP) and the smallest collage system is NP-hard. For computing the smallest RLSLP, we give an encoding in MAX-SAT.

We present an algorithm computing the longest periodic subsequence of a string of length n in O(n7) time with O(n3) space. We obtain improvements when restricting the exponents or extending the search allowing the reported subsequence to be subperiodic down to O(n2) time and O(n) space. By allowing subperiodic subsequences in the output, the task becomes finding the longest bordered subsequence, for which we devise a conditional lower bound.

Since highly repetitive strings, which can be compressed greatly, are ubiquitous nowadays, algorithms working on compressed space have gained much more attention. In particular, converting one compressed format to another without explicit decompression is of great interest as we can take advantage of various compressed formats with small cost of space. In this paper, we consider algorithms to convert run-length encoded Burrows-Wheeler Transform (RLBWT) to Lempel-Ziv 77 (LZ77). We implement a simplified version of the algorithm proposed by Nishimoto and Tabei and propose a two-pass algorithm to reduce its peak memory usage in practice. Experimental results show that the two-pass algorithm uses 23% to 37% less space with up to 10% increase of computational time when we use 8 threads in the second pass.

Computing the matching statistics of patterns with respect to a text is a fundamental task in bioinformatics, but a formidable one when the text is a highly compressed genomic database. Bannai et al. gave an efficient solution for this case, which Rossi et al. recently implemented, but it uses two passes over the patterns and buffers a pointer for each character during the first pass. In this paper, we simplify their solution and make it streaming, at the cost of slowing it down slightly. This means that, first, we can compute the matching statistics of several long patterns (such as whole human chromosomes) in parallel while still using a reasonable amount of RAM; second, we can compute matching statistics online with low latency and thus quickly recognize when a pattern becomes incompressible relative to the database. Our code is available at https://github.com/koeppl/phoni.

Re-Pair is a grammar compression scheme with favorably good compression rates. The computation of Re-Pair comes with the cost of maintaining large frequency tables, which makes it hard to compute Re-Pair on large scale data sets. As a solution for this problem we present, given a text of length n whose characters are drawn from an integer alphabet, an O(n2) time algorithm computing Re-Pair in n lg max(n, τ) bits of working space including the text space, where τ is the number of terminals and non-terminals.

Given a string T of length N, the goal of grammar compression is to construct a small context-free grammar generating only T. Among existing grammar compression methods, RePair (recursive paring) [Larsson and Moffat, 1999] is notable for achieving good compression ratios in practice. In this paper, we propose the first RePair algorithm working in compressed space, i.e., potentially o(N) space for highly compressible texts. The key idea is to give a new way to restructure an arbitrary (context-free) grammar S for T into RePair(T) in compressed space and time. We propose an algorithm for RePair(T) running in O(min(N, nm log N)) space and expected O(min(N, nm log N) m) time or O(min(N, nm log N) log log N) time, where n is the size of S and m is the number of variables in RePair(T). We implemented our O(min(N, nm log N) m)-time algorithm and show it can actually run in compressed space. We also present a new approach to reduce the peak memory usage of existing RePair algorithms combining with our algorithms, and show that the new approach outperforms, both in computation time and space, the most space efficient linear-time RePair implementation to date.

In this paper, we present efficient algorithms for interconversion between Lempel-Ziv 78 (LZ78) encoding and run length encoding (RLE). We show how, given an RLE of size n for a string S, we can compute the corresponding LZ78 encoding of size m for S in O((n + m) log σ) time, where σ is the number of distinct characters appearing in S. We also show how, given an LZ78 encoding of size m for a string S, we can compute the corresponding RLE of size n in O(n + m) time. Both algorithms use O(m) extra working space.

The convolution between a text string S of length N and a pattern string P of length m can be computed in O(N log m) time by FFT. It is known that various types of approximate string matching problems are reducible to convolution. In this paper, we assume that the input text string is given in a compressed form, as a straight-line program (SLP), which is a context free grammar in the Chomsky normal form that derives a single string. Given an SLP S of size n describing a text S of length N, and an uncompressed pattern P of length m, we present a simple O(nm log m)-time algorithm to compute the convolution between S and P. We then show that this can be improved to O(min{nm, N - α} log m) time, where α ≥ 0 is a value that represents the amount of redundancy that the SLP captures with respect to the length-m substrings. The key of the improvement is our new algorithm that computes the convolution between a trie of size r and a pattern string P of length m in O(r log m) time.