On the Complexity of Exact Pattern Matching in Graphs: Binary Strings and Bounded Degree

Exact pattern matching in labeled graphs is the problem of searching paths of a graph $G=(V,E)$ that spell the same string as the pattern $P[1..m]$. This basic problem can be found at the heart of more complex operations on variation graphs in computational biology, of query operations in graph databases, and of analysis operations in heterogeneous networks, where the nodes of some paths must match a sequence of labels or types. We describe a simple conditional lower bound that, for any constant $\epsilon>0$, an $O(|E|^{1 - \epsilon} \, m)$-time or an $O(|E| \, m^{1 - \epsilon})$-time algorithm for exact pattern matching on graphs, with node labels and patterns drawn from a binary alphabet, cannot be achieved unless the Strong Exponential Time Hypothesis (SETH) is false. The result holds even if restricted to undirected graphs of maximum degree three or directed acyclic graphs of maximum sum of indegree and outdegree three. Although a conditional lower bound of this kind can be somehow derived from previous results (Backurs and Indyk, FOCS'16), we give a direct reduction from SETH for dissemination purposes, as the result might interest researchers from several areas, such as computational biology, graph database, and graph mining, as mentioned before. Indeed, as approximate pattern matching on graphs can be solved in $O(|E|\,m)$ time, exact and approximate matching are thus equally hard (quadratic time) on graphs under the SETH assumption. In comparison, the same problems restricted to strings have linear time vs quadratic time solutions, respectively, where the latter ones have a matching SETH lower bound on computing the edit distance of two strings (Backurs and Indyk, STOC'15).Comment: Using Lemma 12 and Lemma 13 might to be enough to prove Lemma 14. However, the proof of Lemma 14 is correct if you assume that the graph used in the reduction is a DAG. Hence, since the problem is already quadratic for a DAG and a binary alphabet, it has to be quadratic also for a general graph and a binary alphabe

On the Complexity of String Matching for Graphs

Peer reviewe

Parameterized Algorithms for String Matching to DAGs: Funnels and Beyond

The problem of String Matching to Labeled Graphs (SMLG) asks to find all the paths in a labeled graph G = (V, E) whose spellings match that of an input string S ? ?^m. SMLG can be solved in quadratic O(m|E|) time [Amir et al., JALG 2000], which was proven to be optimal by a recent lower bound conditioned on SETH [Equi et al., ICALP 2019]. The lower bound states that no strongly subquadratic time algorithm exists, even if restricted to directed acyclic graphs (DAGs). In this work we present the first parameterized algorithms for SMLG on DAGs. Our parameters capture the topological structure of G. All our results are derived from a generalization of the Knuth-Morris-Pratt algorithm [Park and Kim, CPM 1995] optimized to work in time proportional to the number of prefix-incomparable matches. To obtain the parameterization in the topological structure of G, we first study a special class of DAGs called funnels [Millani et al., JCO 2020] and generalize them to k-funnels and the class ST_k. We present several novel characterizations and algorithmic contributions on both funnels and their generalizations

Solving String Problems on Graphs Using the Labeled Direct Product

Suffix trees are an important data structure at the core of optimal solutions to many fundamental string problems, such as exact pattern matching, longest common substring, matching statistics, and longest repeated substring. Recent lines of research focused on extending some of these problems to vertex-labeled graphs, either by using efficient ad-hoc approaches which do not generalize to all input graphs, or by indexing difficult graphs and having worst-case exponential complexities. In the absence of an ubiquitous and polynomial tool like the suffix tree for labeled graphs, we introduce the labeled direct product of two graphs as a general tool for obtaining optimal algorithms in the worst case: we obtain conceptually simpler algorithms for the quadratic problems of string matching (SMLG) and longest common substring (LCSP) in labeled graphs. Our algorithms run in time linear in the size of the labeled product graph, which may be smaller than quadratic for some inputs, and their run-time is predictable, because the size of the labeled direct product graph can be precomputed efficiently. We also solve LCSP on graphs containing cycles, which was left as an open problem by Shimohira et al. in 2011. To show the power of the labeled product graph, we also apply it to solve the matching statistics (MSP) and the longest repeated string (LRSP) problems in labeled graphs. Moreover, we show that our (worst-case quadratic) algorithms are also optimal, conditioned on the Orthogonal Vectors Hypothesis. Finally, we complete the complexity picture around LRSP by studying it on undirected graphs.Peer reviewe

On the Complexity of Exact Pattern Matching in Graphs: Binary Strings and Bounded Degree

International audienc

A Myhill-Nerode Theorem for Generalized Automata, with Applications to Pattern Matching and Compression

The model of generalized automata, introduced by Eilenberg in 1974, allows representing a regular language more concisely than conventional automata by allowing edges to be labeled not only with characters, but also strings. Giammaresi and Montalbano introduced a notion of determinism for generalized automata [STACS 1995]. While generalized deterministic automata retain many properties of conventional deterministic automata, the uniqueness of a minimal generalized deterministic automaton is lost. In the first part of the paper, we show that the lack of uniqueness can be explained by introducing a set $\mathcal{W(A)}$ associated with a generalized automaton $\mathcal{A}$. By fixing $\mathcal{W(A)}$, we are able to derive for the first time a full Myhill-Nerode theorem for generalized automata, which contains the textbook Myhill-Nerode theorem for conventional automata as a degenerate case. In the second part of the paper, we show that the set $\mathcal{W(A)}$ leads to applications for pattern matching and data compression. Wheeler automata [TCS 2017, SODA 2020] are a popular class of automata that can be compactly stored using $e \log \sigma (1 + o(1)) + O(e)$ bits ($e$ being the number of edges, $\sigma$ being the size of the alphabet) in such a way that pattern matching queries can be solved in $\tilde{O}(m)$ time ($m$ being the length of the pattern). In the paper, we show how to extend these results to generalized automata. More precisely, a Wheeler generalized automata can be stored using $\mathfrak{e} \log \sigma (1 + o(1)) + O(e + rn)$ bits so that pattern matching queries can be solved in $\tilde{O}(r m)$ time, where $\mathfrak{e}$ is the total length of all edge labels, $r$ is the maximum length of an edge label and $n$ is the number of states

Long read mapping at scale: Algorithms and applications

Capability to sequence DNA has been around for four decades now, providing ample time to explore its myriad applications and the concomitant development of bioinformatics methods to support them. Nevertheless, disruptive technological changes in sequencing often upend prevailing protocols and characteristics of what can be sequenced, necessitating a new direction of development for bioinformatics algorithms and software. We are now at the cusp of the next revolution in sequencing due to the development of long and ultra-long read sequencing technologies by Pacific Biosciences (PacBio) and Oxford Nanopore Technologies (ONT). Long reads are attractive because they narrow the scale gap between sizes of genomes and sizes of sequenced reads, with the promise of avoiding assembly errors and repeat resolution challenges that plague short read assemblers. However, long reads themselves sport error rates in the vicinity of 10-15%, compared to the high accuracy of short reads (< 1%). There is an urgent need to develop bioinformatics methods to fully realize the potential of long-read sequencers. Mapping and alignment of reads to a reference is typically the first step in genomics applications. Though long read technologies are still evolving, research efforts in bioinformatics have already produced many alignment-based and alignment-free read mapping algorithms. Yet, much work lays ahead in designing provably efficient algorithms, formally characterizing the quality of results, and developing methods that scale to larger input datasets and growing reference databases. While the current model to represent the reference as a collection of linear genomes is still favored due to its simplicity, mapping to graph-based representations, where the graph encodes genetic variations in a human population also becomes imperative. This dissertation work is focused on provably good and scalable algorithms for mapping long reads to both linear and graph references. We make the following contributions: 1. We develop fast and approximate algorithms for end-to-end and split mapping of long reads to reference genomes. Our work is the first to demonstrate scaling to the entire NCBI database, the collection of all curated and non-redundant genomes. 2. We generalize the mapping algorithm to accelerate the related problems of computing pairwise whole-genome comparisons. We shed light on two fundamental biological questions concerning genomic duplications and delineating microbial species boundaries. 3. We provide new complexity results for aligning reads to graphs under Hamming and edit distance models to classify the problem variants for which existence of a polynomial time solution is unlikely. In contrast to prior results that assume alphabets as a function of the problem size, we prove that the problem variants that allow edits in graph remain NP-complete for even constant-sized alphabets, thereby resolving computational complexity of the problem for DNA and protein sequence to graph alignments. 4. Finally, we propose a new parallel algorithm to optimally align long reads to large variation graphs derived from human genomes. It demonstrates near linear scaling on multi-core CPUs, resulting in run-time reduction from multiple days to three hours when aligning a long read set to an MHC human variation graph.Ph.D