Algorithms and Complexity on Indexing Founder Graphs

Equi, Massimo; Norri, Tuukka; Alanko, Jarno; Cazaux, Bastien; Tomescu, Alexandru I.; Mäkinen, Veli

doi:10.1007/s00453-022-01007-w

Cited by 7 publications

(28 citation statements)

References 46 publications

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“…To have a poly-time indexing of G that can solve MEM finding in truly subquadratic time, it is necessary to constrain the family of graphs in question. Therefore, we now focus on Elastic Founder Graphs (EFGs), labeled DAGs supporting poly-time indexing for linear time queries [14]. We will show that the same techniques used to query if Q appears in EFG G can be extended to solve MEM finding on G with arbitrary length threshold κ.…”

Section: Mems With Threshold In Elastic Founder Graphsmentioning

confidence: 99%

“…Equi et al [14] showed that the suffix tree of T 3 can be used to query string Q in G, taking time O(|Q|). We now extend this algorithm to find MEMs between Q and EFG G with threshold κ and spanning more than 3 nodes.…”

Section: A Co-linear Chaining On Strings Using Mems Gives Lcsmentioning

confidence: 99%

“…Due to the conditional hardness result previously mentioned, a truly sub-quadratic time solution is not likely to be found, even when given an arbitrary threshold (lower bound) for MEM length. Therefore, we limit our attention to indexable elastic founder graphs [14]. We show that, if G is an indexable elastic founder graph, we can report all κ-MEMs (MEMs of length at least κ) in time O(nH 2 + m + M κ ), where H (height), H ≤ k, is the maximum number of nodes in a block of the graph, and M κ ≤ M 1 is the number of κ-MEMs.…”

Section: Introductionmentioning

confidence: 99%

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Chaining of Maximal Exact Matches in Graphs

Rizzo¹,

Cáceres²,

Mäkinen³

2023

Preprint

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We study the problem of finding maximal exact matches (MEMs) between a query string Q and a labeled directed acyclic graph (DAG) G = (V, E, ℓ) and subsequently co-linearly chaining these matches. We show that it suffices to compute MEMs between node labels and Q (node MEMs) to encode full MEMs. Node MEMs can be computed in linear time and we show how to co-linearly chain them to solve the Longest Common Subsequence (LCS) problem between Q and G. Our chaining algorithm is the first to consider a symmetric formulation of the chaining problem in graphs and runs inwhere k is the width (minimum number of paths covering the nodes) of G, and N is the number of node MEMs.We then consider the problem of finding MEMs when the input graph is an indexable elastic founder graph (subclass of labeled DAGs studied by Equi et al., Algorithmica 2022). For arbitrary input graphs, the problem cannot be solved in truly sub-quadratic time under SETH (Equi et al., ICALP 2019). We show that we can report all MEMs between Q and an indexable elastic founder graph in time O(nH 2 + m + Mκ), where n is the total length of node labels, H is the maximum number of nodes in a block of the graph, m = |Q|, and Mκ is the number of MEMs of length at least κ.The results extend to the indexing problem, where the graph is preprocessed and a set of queries is processed as a batch.

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Section: Mems With Threshold In Elastic Founder Graphsmentioning

confidence: 99%

Section: A Co-linear Chaining On Strings Using Mems Gives Lcsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Chaining of Maximal Exact Matches in Graphs

Rizzo¹,

Cáceres²,

Mäkinen³

2023

Preprint

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“…The idea behind founder graphs is that a multiple alignment of few founder sequences can be used to approximate the input MSA, with the feature that each row of the MSA is a recombination of the founders. Unlike ED strings, that are believed not to be efficiently indexable [26] (and indeed their value is to enable fast on-line string matching algorithms), some subclasses of founder graphs are, and a recent line of research is devoted to constructing and indexing such structures [32,20]. Like founder graphs, ED strings support the recombination of different rows of the MSA between consecutive columns.…”

Section: Other Related Workmentioning

confidence: 99%

Elastic-Degenerate String Matching via Fast Matrix Multiplication

Bernardini¹,

Gawrychowski²,

Pisanti³

et al. 2022

SIAM J. Comput.

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An elastic-degenerate (ED) string is a sequence of n finite sets of strings of total length N , introduced to represent a set of related DNA sequences, also known as a pangenome. The ED string matching (EDSM) problem consists in reporting all occurrences of a pattern of length m in an ED text. The EDSM problem has recently received some attention by the combinatorial pattern matching community, culminating in an Õ(nm ω−1 ) + O(N )-time algorithm [Bernardini et al., SIAM J. Comput. 2022], where ω denotes the matrix multiplication exponent and the Õ(•) notation suppresses polylog factors. In the k-EDSM problem, the approximate version of EDSM, we are asked to report all pattern occurrences with at most k errors. k-EDSM can be solved in O(k 2 mG + kN ) time, under edit distance, or O(kmG + kN ) time, under Hamming distance, where G denotes the total number of strings in the ED text [Bernardini et al., Theor. Comput. Sci. 2020]. Unfortunately, G is only bounded by N , and so even for k = 1, the existing algorithms run in Ω(mN ) time in the worst case. In this paper we make progress in this direction. We show that 1-EDSM can be solved in O((nm 2 + N ) log m) or O(nm 3 + N ) time under edit distance. For the decision version of the problem, we present a faster O(nm 2 √ log m + N log log m)-time algorithm. We also show that 1-EDSM can be solved in O(nm 2 + N log m) time under Hamming distance. Our algorithms for edit distance rely on non-trivial reductions from 1-EDSM to special instances of classic computational geometry problems (2d rectangle stabbing or 2d range emptiness), which we show how to solve efficiently. In order to obtain an even faster algorithm for Hamming distance, we rely on employing and adapting the k-errata trees for indexing with errors [Cole et al., STOC 2004].

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“…Due to the difficulty of string search in general graphs, Equi et al [14] studied graphs obtained from multiple sequence alignments (MSAs), where an MSA[1..m, 1..n] is a matrix composed of m aligned rows that are strings of length n, drawn from an alphabet Σ plus a special gap symbol −. As we describe in Section 2, any segmentation of an MSA naturally induces a graph consisting of labeled nodes, partitioned into blocks.…”

Section: Introductionmentioning

confidence: 99%

Elastic Founder Graphs Improved and Enhanced

Rizzo¹,

Equi²,

Norri³

et al. 2023

Preprint

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Indexing labeled graphs for pattern matching is a central challenge of pangenomics. Equi et al. (Algorithmica, 2022) developed the Elastic Founder Graph (EFG) representing an alignment of m sequences of length n, drawn from alphabet Σ plus the special gap character: the paths spell the original sequences or their recombination. By enforcing the semi-repeat-free property, the EFG admits a polynomial-space index for linear-time pattern matching, breaking through the conditional lower bounds on indexing labeled graphs (Equi et al., SOFSEM 2021). In this work we improve the space of the EFG index answering pattern matching queries in linear time, from linear in the length of all strings spelled by three consecutive node labels, to linear in the size of the edge labels. Then, we develop linear-time construction algorithms optimizing for different metrics: we improve the existing linearithmic construction algorithms to O(mn), by solving the novel exclusive ancestor set problem on trees; we propose, for the simplified gapless setting, an O(mn)-time solution minimizing the maximum block height, that we generalize by substituting block height with prefix-aware height. Finally, to show the versatility of the framework, we develop a BWT-based EFG index and study how to encode and perform document listing queries on a set of paths of the graphs, reporting which paths present a given pattern as a substring. We propose the EFG framework as an improved and enhanced version of the framework for the gapless setting, along with construction methods that are valid in any setting concerned with the segmentation of aligned sequences.

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Algorithms and Complexity on Indexing Founder Graphs

Cited by 7 publications

References 46 publications

Chaining of Maximal Exact Matches in Graphs

Chaining of Maximal Exact Matches in Graphs

Elastic-Degenerate String Matching via Fast Matrix Multiplication

Elastic Founder Graphs Improved and Enhanced

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