In this paper, we present a new algorithm for calculating exact hypervolumes. Given a set of d -dimensional points, it computes the hypervolume of the dominated space. Determining this value is an important subroutine of multiobjective evolutionary algorithms. We analyze the quick hypervolume (QHV) algorithm theoretically and experimentally. The theoretical results are a significant contribution to the current state of the art. Moreover, the experimental performance is also very competitive, compared with existing exact hypervolume algorithms.

Chaos Game Representation (CGR) is an iterated function that bijectively maps discrete sequences into a continuous domain. As a result, discrete sequences can be object of statistical and topological analyses otherwise reserved to numerical systems. Characteristically, CGR coordinates of substrings sharing an L-long suffix will be located within 2 -L distance of each other. In the two decades since its original proposal, CGR has been generalized beyond its original focus on genomic sequences and has been successfully applied to a wide range of problems in bioinformatics. This report explores the possibility that it can be further extended to approach algorithms that rely on discrete, graph-based representations.

Suffix trees are by far the most important data structure in stringology, with a myriad of applications in fields like bioinformatics and information retrieval. Classical representations of suffix trees require Θ( n log n ) bits of space, for a string of size n . This is considerably more than the n log ₂ σ bits needed for the string itself, where σ is the alphabet size. The size of suffix trees has been a barrier to their wider adoption in practice. Recent compressed suffix tree representations require just the space of the compressed string plus Θ( n ) extra bits. This is already spectacular, but the linear extra bits are still unsatisfactory when σ is small as in DNA sequences. In this article, we introduce the first compressed suffix tree representation that breaks this Θ( n )-bit space barrier. The Fully Compressed Suffix Tree (FCST) representation requires only sublinear space on top of the compressed text size, and supports a wide set of navigational operations in almost logarithmic time. This includes extracting arbitrary text substrings, so the FCST replaces the text using almost the same space as the compressed text. An essential ingredient of FCSTs is the lowest common ancestor (LCA) operation. We reveal important connections between LCAs and suffix tree navigation. We also describe how to make FCSTs dynamic, that is, support updates to the text. The dynamic FCST also supports several operations. In particular, it can build the static FCST within optimal space and polylogarithmic time per symbol. Our theoretical results are also validated experimentally, showing that FCSTs are very effective in practice as well.

A compressed full-text self-index for a text T is a data structure requiring reduced space and able to search for patterns P in T. It can also reproduce any substring of T, thus actually replacing T. Despite the recent explosion of interest on compressed indexes, there has not been much progress on functionalities beyond the basic exact search. In this paper we focus on indexed approximate string matching (ASM), which is of great interest, say, in bioinformatics. We study ASM algorithms for Lempel-Ziv compressed indexes and for compressed suffix trees/arrays. Most compressed self-indexes belong to one of these classes. We start by adapting the classical method of partitioning into exact search to self-indexes, and optimize it over a representative of either class of self-index. Then, we show that a Lempel- Ziv index can be seen as an extension of the classical q-samples index. We give new insights on this type of index, which can be of independent interest, and then apply them to a Lempel- Ziv index. Finally, we improve hierarchical verification, a successful technique for sequential searching, so as to extend the matches of pattern pieces to the left or right. Most compressed suffix trees/arrays support the required bidirectionality, thus enabling the implementation of the improved technique. In turn, the improved verification largely reduces the accesses to the text, which are expensive in self-indexes. We show experimentally that our algorithms are competitive and provide useful space-time tradeoffs compared to classical indexes.