On the Complexity of String Matching for Graphs

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

doi:10.4230/lipics.icalp.2019.55

Cited by 19 publications

(31 citation statements)

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“…For Problem 2 we provide a hardness result based on SETH, which is frequently used for establishing conditional optimality of polynomial time algorithms [1,7,13,18,19,24]. We refer the reader to [43] for the definition of SETH and for the reduction to the Orthogonal Vectors problem (OV), which is utilized to prove Theorem 2.…”

Section: Technical Background and Our Resultsmentioning

confidence: 99%

“…(this paper) NO Strongly Sub-O(|E|m) Alg. NP-Complete • General graphs [13,19] • General graphs: (including DAGs with Substitutions/Edits to vertex labels [6,25] Hard total degree ≤ 3)…”

Section: Exact Matching Approximate Matching Solvable In Linear Timementioning

confidence: 99%

“…For general graphs, we can consider exact and approximate matching. For exact matching, conditional lower-bounds based on the Strong Exponential Time Hypothesis (SETH), and other conjectures in circuit complexity, indicate that an O(|E|m 1−ε + |E| 1−ε m) time algorithm with any constant ε > 0, for a graph with |E| edges and a pattern of length m, is highly unlikely (as is the ability to shave more than a constant number of logarithmic factors from the O(|E|m) time complexity) [13,19]. These results hold for even very restricted types of graphs, for example, DAGs with maximum total degree three and binary alphabets.…”

Section: Exact Matching Approximate Matching Solvable In Linear Timementioning

confidence: 99%

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The Complexity of Approximate Pattern Matching on De Bruijn Graphs

Gibney¹,

Thankachan²,

Aluru³

2022

Preprint

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Aligning a sequence to a walk in a labeled graph is a problem of fundamental importance to Computational Biology. For finding a walk in an arbitrary graph with |E| edges that exactly matches a pattern of length m, a lower bound based on the Strong Exponential Time Hypothesis (SETH) implies an algorithm significantly faster than O(|E|m) time is unlikely [Equi et al., ICALP 2019]. However, for many special graphs, such as de Bruijn graphs, the problem can be solved in linear time [Bowe et al., WABI 2012]. For approximate matching, the picture is more complex. When edits (substitutions, insertions, and deletions) are only allowed to the pattern, or when the graph is acyclic, the problem is again solvable in O(|E|m) time. When edits are allowed to arbitrary cyclic graphs, the problem becomes NP-complete, even on binary alphabets [Jain et al., RECOMB 2019]. These results hold even when edits are restricted to only substitutions. Despite the popularity of de Bruijn graphs in Computational Biology, the complexity of approximate pattern matching on de Bruijn graphs remained open. We investigate this problem and show that the properties that make de Bruijn graphs amenable to efficient exact pattern matching do not extend to approximate matching, even when restricted to the substitutions only case with alphabet size four. Specifically, we prove that determining the existence of a matching walk in a de Bruijn graph is NP-complete when substitutions are allowed to the graph. In addition, we demonstrate that an algorithm significantly faster than O(|E|m) is unlikely for de Bruijn graphs in the case where only substitutions are allowed to the pattern. This stands in contrast to pattern-to-text matching where exact matching is solvable in linear time, like on de Bruijn graphs, but approximate matching under substitutions is solvable in subquadratic Õ(n √ m) time, where n is the text's length [Abrahamson, SIAM J. Computing 1987].

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Section: Technical Background and Our Resultsmentioning

confidence: 99%

Section: Exact Matching Approximate Matching Solvable In Linear Timementioning

confidence: 99%

Section: Exact Matching Approximate Matching Solvable In Linear Timementioning

confidence: 99%

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The Complexity of Approximate Pattern Matching on De Bruijn Graphs

Gibney¹,

Thankachan²,

Aluru³

2022

Preprint

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“…Before diving into compressed indexes for labeled graphs, we spend a paragraph on the complexity of matching strings on labeled graphs. The main results in this direction are due to Backurs and Indyk [51] and to Equi et al [52][53][54], and considered the on-line version of the problem: both pre-processing and query time are counted towards the total running time. The former work [51] proved that, unless the Strong Exponential Time Hypothesis (SETH) [55] is false, in the worst case no algorithm can match a regular expression of size e against a string of length m in time O((m • e) 1−δ ), for any constant δ > 0.…”

Section: Conditional Lower Boundsmentioning

confidence: 99%

Subpath Queries on Compressed Graphs: a Survey

Prezza¹

2020

Preprint

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Text indexing is a classical algorithmic problem that has been studied for over four decades: given a text T, pre-process it off-line so that, later, we can quickly count and locate the occurrences of any string (the query pattern) in T in time proportional to the query's length. The earliest optimal-time solution to the problem, the suffix tree, dates back to 1973 and requires up to two orders of magnitude more space than the plain text just to be stored. In the year 2000, two breakthrough works showed that efficient queries can be achieved without this space overhead: a fast index be stored in a space proportional to the text's entropy. These contributions had an enormous impact in bioinformatics: nowadays, virtually any DNA aligner employs compressed indexes. Recent trends considered more powerful compression schemes (dictionary compressors) and generalizations of the problem to labeled graphs: after all, texts can be viewed as labeled directed paths. In turn, since finite state automata can be considered as a particular case of labeled graphs, these findings created a bridge between the fields of compressed indexing and regular language theory, ultimately allowing to index regular languages and promising to shed new light on problems such as regular expression matching. This survey is a gentle introduction to the main landmarks of the fascinating journey that took us from suffix trees to today's compressed indexes for labeled graphs and regular languages.

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“…When L is represented as an NFA (equivalently, a regular expression) of size m, existing on-line algorithms [3] solve the problem in O(πm) time, π being the length of the query pattern. Recent lower bounds by Backurs and Indyk [4], Equi et al [8,7], Potechin and Shallit [16], and Gibney [11] show that, unless important conjectures such as the Strong Exponential Time Hypothesis (SETH) [12] fail, this complexity cannot be significantly improved. This holds even in the off-line setting (the subject matter of our work) where L can be pre-processed in an index in polynomial time and the complexity is measured in terms of query times [9].…”

Section: Introductionmentioning

confidence: 99%

Which Regular Languages can be Efficiently Indexed?

Cotumaccio¹,

D’Agostino²,

Policriti³

et al. 2021

Preprint

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Consider the problem of matching a pattern P of length π against the elements of a given regular language L in the setting where L can be pre-processed off-line in a fast data structure (an index). Regular expression matching is an ubiquitous problem in computer science, finding fundamental applications in areas including, but not limited to, natural language processing, search engines, compilers, and databases. Recent results have settled the exact complexity of this problem: Θ(πm) time is necessary and sufficient for indexed pattern matching queries, where m is the size of an NFA recognizing L. This, however, does not mean that all regular languages are hard to index: for instance, for the sub-class of Wheeler languages [SODA'20] we can reduce query time to the optimal O(π). A Wheeler language admits a total order of a finite refinement of its Myhill-Nerode equivalence classes reflecting the co-lexicographic order of their elements. This boosts indexing performance because classes whose elements are suffixed by P form a range in this order. In [SODA'21], this technique was extended to arbitrary NFAs by allowing the order to be partial. This line of attack suggested that the width p of such an order is the parameter ultimately capturing the fine-grained complexity of the problem: (i) indexed pattern matching can always be solved in O(πp 2 ) time, (ii) the standard powerset construction algorithm always produces an output whose size is exponential in p rather than in the input's size, and (iii) p even determines how succinctly NFAs can be encoded.In the present work, we take a step further and study the hierarchy of p-sortable languages: regular languages accepted by automata of width p. Our main contributions are the following: (i) we show that the hierarchy is strict and does not collapse, (ii) we provide (exponential) upper and lower bounds relating the minimum widths of equivalent NFAs and DFAs, and (iii) we characterize DFAs of minimum p for a given L via a co-lexicographic variant of the Myhill-Nerode theorem. Our findings imply that in polynomial time we can build an index breaking the worst-case conditional lower bound of Ω(πm), whenever the input NFA's width is at most ϵ log 2 m, for any constant 0 ≤ ϵ < 1/2.

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On the Complexity of String Matching for Graphs

Cited by 19 publications

References 0 publications

The Complexity of Approximate Pattern Matching on De Bruijn Graphs

The Complexity of Approximate Pattern Matching on De Bruijn Graphs

Subpath Queries on Compressed Graphs: a Survey

Which Regular Languages can be Efficiently Indexed?

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