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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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On the Complexity of Sequence to Graph Alignment
Availability of extensive genetics data across multiple individuals and populations is driving the growing importance of graph based reference representations. Aligning sequences to graphs is a fundamental operation on several types of sequence graphs (variation graphs, assembly graphs, pan-genomes, etc.) and their biological applications. Though research on sequence to graph alignments is nascent, it can draw from related work on pattern matching in hypertext. In this paper, we study sequence to graph alignment problems under Hamming and edit distance models, and linear and affine gap penalty functions, for multiple variants of the problem that allow changes in query alone, graph alone, or in both. We prove that when changes are permitted in graphs either standalone or in conjunction with changes in the query, the sequence to graph alignment problem is NP -complete under both Hamming and edit distance models for alphabets of size ≥2 . For the case where only changes to the sequence are permitted, we present an O(|V|+m|E|) time algorithm, where m denotes the query size, and V and E denote the vertex and edge sets of the graph, respectively. Our result is generalizable to both linear and affine gap penalty functions, and improves upon the run-time complexity of existing algorithms.
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- Award ID(s):
- 1816027
- PAR ID:
- 10122205
- Date Published:
- Journal Name:
- Lecture Notes in Computer Science
- Volume:
- 11467
- Page Range / eLocation ID:
- 85-100
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1007/978-3-030-17083-7
