Graphs Cannot Be Indexed in Polynomial Time for Sub-quadratic Time String Matching, Unless SETH Fails

Equi, Massimo; Mäkinen, Veli; Tomescu, Alexandru I.

doi:10.1007/978-3-030-67731-2_44

Cited by 24 publications

(19 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As the reader can verify, the presented construction is a linear independent component (LIC) reduction [20], and hence it follows that one cannot index generic elastic founder graphs in polynomial time for efficient string search, unless the Orthogonal Vector Hypothesis or the Strong Exponential Time Hypothesis (SETH) are false [20]. Thus, a property like repeatfreeness is required.…”

Section: Hardnessmentioning

confidence: 97%

“…Since it is NP-hard to recognize if a given graph is Wheeler [17], it is of interest to look for other graph classes that could provide some indexability functionality. Unfortunately, quite simple graphs turn out to be hard to index [19,20] (under the Strong Exponential Time Hypothesis). But as we will see later, further restrictions on graphs change the situation: We show that there exists a family of founder graphs that can be indexed in linear time to support linear time queries.…”

Section: Related Workmentioning

confidence: 99%

See 1 more Smart Citation

Algorithms and Complexity on Indexing Founder Graphs

Equi¹,

Norri²,

Alanko³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

We introduce a compact pangenome representation based on an optimal segmentation concept that aims to reconstruct founder sequences from a multiple sequence alignment (MSA). Such founder sequences have the feature that each row of the MSA is a recombination of the founders. Several linear time dynamic programming algorithms have been previously devised to optimize segmentations that induce founder blocks that then can be concatenated into a set of founder sequences. All possible concatenation orders can be expressed as a founder graph. We observe a key property of such graphs: if the node labels (founder segments) do not repeat in the paths of the graph, such graphs can be indexed for efficient string matching. We call such graphs repeat-free founder graphs when constructed from a gapless MSA and repeat-free elastic founder graphs when constructed from a general MSA with gaps.We give a linear time algorithm and a parameterized near linear time algorithm to construct a repeat-free founder graph and a repeat-free elastic founder graph, respectively. The two algorithms combine techniques from the founder segmentation algorithms (Cazaux et al., SPIRE 2019) and fully-functional bidirectional Burrows-Wheeler index (Belazzougui and Cunial, CPM 2019). We derive a tailored succinct index structure to support queries of arbitrary length in the paths of a repeat-free (elastic) founder graph. In addition, we show how to turn a repeat-free (elastic) founder graph into a Wheeler graph (Gagie et al., TCS 2017) in polynomial time. This reduction enables the use the versatile machinery (Alanko et al., SODA 2020) developed for Wheeler graphs.Furthermore, we show that a property such as repeat-freeness is essential for indexability. In particular, we show that unless the Strong Exponential Time Hypothesis (SETH) fails, one cannot build an index on an elastic founder graph in polynomial time to support fast queries.

show abstract

Section: Hardnessmentioning

confidence: 97%

Section: Related Workmentioning

confidence: 99%

Algorithms and Complexity on Indexing Founder Graphs

Equi¹,

Norri²,

Alanko³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…It turns out that this kind of automata, called Wheeler automata, (a) admit an efficient index data structure for searching subpaths labeled with a given query pattern, and (b) enable a representation of the graph in a space proportional to that of the edges' labels since the topology can be encoded with just O(1) bits per node [13] (as well as enabling more advanced compression mechanisms, see [3,11]). This is in contrast with the fact that general graphs require a logarithmic (in the graph's size) number of bits per edge to be represented, as well as with recent results showing that in general, the subpath search problem can not be solved in subquadratic time, unless the strong exponential time hypothesis is false [4,5,6,7,10]. A WDFAs A recognizing the language L d = ac * + dc * f .…”

Section: Introductionmentioning

confidence: 99%

Ordering regular languages: a danger zone

D’Agostino¹,

Martincigh²,

Policriti³

2021

Preprint

View full text Add to dashboard Cite

Ordering the collection of states of a given automaton starting from an order of the underlying alphabet is a natural move towards a computational treatment of the language accepted by the automaton. Along this path, Wheeler graphs have been recently introduced as an extension/adaptation of the Burrows-Wheeler Transform (the now famous BWT, originally defined on strings) to graphs. These graphs constitute an important data-structure for languages, since they allow a very efficient storage mechanism for the transition function of an automaton, while providing a fast support to all sorts of substring queries. This is possible as a consequence of a property-the so-called path coherence-valid on Wheeler graphs and consisting in an ordering on nodes that "propagates" to (collections of) strings. By looking at a Wheeler graph as an automaton, the ordering on strings corresponds to the co-lexicographic order of the words entering each state. This leads naturally to consider the class of regular languages accepted by Wheeler automata, i.e. the Wheeler languages. It has been shown that, as opposed to the general case, the classic determinization by powerset construction is polynomial on Wheeler languages. As a consequence, most of the classical problems turn out to be "easy"that is, solvable in polynomial time-on Wheeler languages. Moreover, deciding whether a DFA is Wheeler and deciding whether a DFA accepts a Wheeler language is polynomial. Our contribution here is to put an upper bound to easy problems. For instance, whenever we generalize by switching to general NFAs or by not fixing an order of the underlying alphabet, the above mentioned problems become "hard"-that is NP-complete or even PSPACE-complete.

show abstract

“…This lower bound holds even if requiring only exact matches, the graph is acyclic, i.e. a DAG [12, 15], and we allow any polynomial-time indexing of the graph [13].…”

Section: Introductionmentioning

confidence: 99%

Chaining for Accurate Alignment of Erroneous Long Reads to Acyclic Variation Graphs^*

Cáceres

Salmela

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

Aligning reads to a variation graph is a standard task in pangenomics, with downstream applications in e.g., improving variant calling. While the vg toolkit (Garrison et al., Nature Biotechnology, 2018) is a popular aligner of short reads, GraphAligner (Rautiainen and Marschall, Genome Biology, 2020) is the state-of-the-art aligner of long reads. GraphAligner works by finding candidate read occurrences based on individually extending the best seeds of the read in the variation graph. However, a more principled approach recognized in the community is to co-linearly chain multiple seeds. We present a new algorithm to co-linearly chain a set of seeds in an acyclic variation graph, together with the first efficient implementation of such a co-linear chaining algorithm into a new aligner of long reads to variation graphs, GraphChainer. Compared to GraphAligner, at a normalized edit distance threshold of 40%, it aligns 9% to 12% more reads, and 15% to 19% more total read length, on real PacBio reads from human chromosomes 1 and 22. On both simulated and real data, GraphChainer aligns between 97% and 99% of all reads, and of total read length. At the more stringent normalized edit distance threshold of 30%, GraphChainer aligns up to 29% more total real read length than GraphAligner. GraphChainer is freely available at https://github.com/algbio/GraphChainer

show abstract

Graphs Cannot Be Indexed in Polynomial Time for Sub-quadratic Time String Matching, Unless SETH Fails

Cited by 24 publications

References 31 publications

Algorithms and Complexity on Indexing Founder Graphs

Algorithms and Complexity on Indexing Founder Graphs

Ordering regular languages: a danger zone

Chaining for Accurate Alignment of Erroneous Long Reads to Acyclic Variation Graphs^*

Contact Info

Product

Resources

About

Graphs Cannot Be Indexed in Polynomial Time for Sub-quadratic Time String Matching, Unless SETH Fails

Cited by 24 publications

References 31 publications

Algorithms and Complexity on Indexing Founder Graphs

Algorithms and Complexity on Indexing Founder Graphs

Ordering regular languages: a danger zone

Chaining for Accurate Alignment of Erroneous Long Reads to Acyclic Variation Graphs*

Contact Info

Product

Resources

About

Chaining for Accurate Alignment of Erroneous Long Reads to Acyclic Variation Graphs^*