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1. Efficient Parallel Output-Sensitive Edit Distance

   https://doi.org/10.4230/LIPIcs.ESA.2023.40  

   Ding, Xiangyun; Dong, Xiaojun; Gu, Yan; Liu, Youzhe; Sun, Yihan (September 2023, 31st Annual European Symposium on Algorithms (ESA 2023))

   Gørtz, Inge Li; Farach-Colton, Martin; Puglisi, Simon J.; Herman, Grzegorz (Ed.)

   In this paper, we study efficient parallel edit distance algorithms, both in theory and in practice. Given two strings A[1..n] and B[1..m], and a set of operations allowed to edit the strings, the edit distance between A and B is the minimum number of operations required to transform A into B. In this paper, we use edit distance to refer to the Levenshtein distance, which allows for unit-cost single-character edits (insertions, deletions, substitutions). Sequentially, a standard Dynamic Programming (DP) algorithm solves edit distance with Θ(nm) cost. In many real-world applications, the strings to be compared are similar to each other and have small edit distances. To achieve highly practical implementations, we focus on output-sensitive parallel edit-distance algorithms, i.e., to achieve asymptotically better cost bounds than the standard Θ(nm) algorithm when the edit distance is small. We study four algorithms in the paper, including three algorithms based on Breadth-First Search (BFS), and one algorithm based on Divide-and-Conquer (DaC). Our BFS-based solution is based on the Landau-Vishkin algorithm. We implement three different data structures for the longest common prefix (LCP) queries needed in the algorithm: the classic solution using parallel suffix array, and two hash-based solutions proposed in this paper. Our DaC-based solution is inspired by the output-insensitive solution proposed by Apostolico et al., and we propose a non-trivial adaption to make it output-sensitive. All of the algorithms studied in this paper have good theoretical guarantees, and they achieve different tradeoffs between work (total number of operations), span (longest dependence chain in the computation), and space. We test and compare our algorithms on both synthetic data and real-world data, including DNA sequences, Wikipedia texts, GitHub repositories, etc. Our BFS-based algorithms outperform the existing parallel edit-distance implementation in ParlayLib in all test cases. On cases with fewer than 10⁵ edits, our algorithm can process input sequences of size 10⁹ in about ten seconds, while ParlayLib can only process sequences of sizes up to 10⁶ in the same amount of time. By comparing our algorithms, we also provide a better understanding of the choice of algorithms for different input patterns. We believe that our paper is the first systematic study in the theory and practice of parallel edit distance. 

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2. Levenshtein graphs: Resolvability, automorphisms & determining sets

   https://doi.org/10.1016/j.disc.2022.113310  

   Ruth, Perrin E.; Lladser, Manuel E. (May 2023, Discrete Mathematics)

   We introduce the notion of Levenshtein graphs, an analog to Hamming graphs but using the edit distance instead of the Hamming distance; in particular, vertices in Levenshtein graphs may be strings (i.e., words or sequences of characters in a reference alphabet) of possibly different lengths. We study various properties of these graphs, including a necessary and sufficient condition for their shortest path distance to be identical to the edit distance, and characterize their automorphism group and determining number. We also bound the metric dimension (i.e. minimum resolving set size) of Levenshtein graphs. Regarding the latter, recall that a run is a string composed of identical characters. We construct a resolving set of two-run strings and an algorithm that computes the edit distance between a string of length k and any single-run or two-run string in operations. 

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3. The effect of genome graph expressiveness on the discrepancy between genome graph distance and string set distance

   Qiu, Yutong; Kingsford, Carl (January 2022, ISCB ISMB 2022)

   Motivation: Intra-sample heterogeneity describes the phenomenon where a genomic sample contains a diverse set of genomic sequences. In practice, the true string sets in a sample are often unknown due to limitations in sequencing technology. In order to compare heterogeneous samples, genome graphs can be used to represent such sets of strings. However, a genome graph is generally able to represent a string set universe that contains multiple sets of strings in addition to the true string set. This difference between genome graphs and string sets is not well characterized. As a result, a distance metric between genome graphs may not match the distance between true string sets. Results: We extend a genome graph distance metric, Graph Traversal Edit Distance (GTED) proposed by Ebrahimpour Boroojeny et al., to FGTED to model the distance between heterogeneous string sets and show that GTED and FGTED always underestimate the Earth Mover’s Edit Distance (EMED) between string sets. We introduce the notion of string set universe diameter of a genome graph. Using the diameter, we are able to upper-bound the deviation of FGTED from EMED and to improve FGTED so that it reduces the average error in empirically estimating the similarity between true string sets. On simulated T-cell receptor sequences and actual Hepatitis B virus genomes, we show that the diameter-corrected FGTED reduces the average deviation of the estimated distance from the true string set distances by more than 250%. Availability and implementation: Data and source code for reproducing the experiments are available at: https:// github.com/Kingsford-Group/gtedemedtest/. 

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4. SecureED: Secure Multiparty Edit Distance for Genomic Sequences

   https://doi.org/10.56553/popets-2025-0072  

   Gao, Jiahui; Palanikumar, Yagaagowtham; Mouris, Dimitris; Nguyen, Duong; Trieu, Ni (April 2025, Proceedings on Privacy Enhancing Technologies)

   DNA edit distance (ED) measures the minimum number of single nucleotide insertions, substitutions, or deletions required to convert a DNA sequence into another. ED has broad applications in healthcare such as sequence alignment, genome assembly, functional annotation, and drug discovery. Privacy-preserving computation is essential in this context to protect sensitive genomic data. Nonetheless, the existing secure DNA edit distance solutions lack efficiency when handling large data sequences or resort to approximations and fail to accurately compute the metric. In this work, we introduce ScureED, a protocol that tackles these limitations, resulting in a significant performance enhancement of approximately 2-24 times compared to existing methods. Our protocol computes a secure ED between two genomes, each comprising 1,000 letters, in just a few seconds. The underlying technique of our protocol is a novel approach that transforms the established approximate matching technique (i.e., the Ukkonen algorithm) into exact matching, exploiting the inherent similarity in human DNA to achieve cost-effectiveness. Furthermore, we introduce various optimizations tailored for secure computation in scenarios with a limited input domain, such as DNA sequences composed solely of the four nucleotide letters. 

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5. 3GOLD: optimized Levenshtein distance for clustering third-generation sequencing data

   https://doi.org/10.1186/s12859-022-04637-7  

   Logan, Robert; Fleischmann, Zoe; Annis, Sofia; Wehe, Amy Wangsness; Tilly, Jonathan L.; Woods, Dori C.; Khrapko, Konstantin (December 2022, BMC Bioinformatics)

   Abstract Background Third-generation sequencing offers some advantages over next-generation sequencing predecessors, but with the caveat of harboring a much higher error rate. Clustering-related sequences is an essential task in modern biology. To accurately cluster sequences rich in errors, error type and frequency need to be accounted for. Levenshtein distance is a well-established mathematical algorithm for measuring the edit distance between words and can specifically weight insertions, deletions and substitutions. However, there are drawbacks to using Levenshtein distance in a biological context and hence has rarely been used for this purpose. We present novel modifications to the Levenshtein distance algorithm to optimize it for clustering error-rich biological sequencing data. Results We successfully introduced a bidirectional frameshift allowance with end-user determined accommodation caps combined with weighted error discrimination. Furthermore, our modifications dramatically improved the computational speed of Levenstein distance. For simulated ONT MinION and PacBio Sequel datasets, the average clustering sensitivity for 3GOLD was 41.45% (S.D. 10.39) higher than Sequence-Levenstein distance, 52.14% (S.D. 9.43) higher than Levenshtein distance, 55.93% (S.D. 8.67) higher than Starcode, 42.68% (S.D. 8.09) higher than CD-HIT-EST and 61.49% (S.D. 7.81) higher than DNACLUST. For biological ONT MinION data, 3GOLD clustering sensitivity was 27.99% higher than Sequence-Levenstein distance, 52.76% higher than Levenshtein distance, 56.39% higher than Starcode, 48% higher than CD-HIT-EST and 70.4% higher than DNACLUST. Conclusion Our modifications to Levenshtein distance have improved its speed and accuracy compared to the classic Levenshtein distance, Sequence-Levenshtein distance and other commonly used clustering approaches on simulated and biological third-generation sequenced datasets. Our clustering approach is appropriate for datasets of unknown cluster centroids, such as those generated with unique molecular identifiers as well as known centroids such as barcoded datasets. A strength of our approach is high accuracy in resolving small clusters and mitigating the number of singletons. 

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