510 research outputs found
Faster Approximate String Matching for Short Patterns
We study the classical approximate string matching problem, that is, given
strings and and an error threshold , find all ending positions of
substrings of whose edit distance to is at most . Let and
have lengths and , respectively. On a standard unit-cost word RAM with
word size we present an algorithm using time When is
short, namely, or this
improves the previously best known time bounds for the problem. The result is
achieved using a novel implementation of the Landau-Vishkin algorithm based on
tabulation and word-level parallelism.Comment: To appear in Theory of Computing System
Combinatorial Algorithms for Subsequence Matching: A Survey
In this paper we provide an overview of a series of recent results regarding
algorithms for searching for subsequences in words or for the analysis of the
sets of subsequences occurring in a word.Comment: This is a revised version of the paper with the same title which
appeared in the Proceedings of NCMA 2022, EPTCS 367, 2022, pp. 11-27 (DOI:
10.4204/EPTCS.367.2). The revision consists in citing a series of relevant
references which were not covered in the initial version, and commenting on
how they relate to the results we survey. arXiv admin note: text overlap with
arXiv:2206.1389
Trace Reconstruction: Generalized and Parameterized
In the beautifully simple-to-state problem of trace reconstruction, the goal is to reconstruct an unknown binary string x given random "traces" of x where each trace is generated by deleting each coordinate of x independently with probability p<1. The problem is well studied both when the unknown string is arbitrary and when it is chosen uniformly at random. For both settings, there is still an exponential gap between upper and lower sample complexity bounds and our understanding of the problem is still surprisingly limited. In this paper, we consider natural parameterizations and generalizations of this problem in an effort to attain a deeper and more comprehensive understanding. Perhaps our most surprising results are:
1) We prove that exp(O(n^(1/4) sqrt{log n})) traces suffice for reconstructing arbitrary matrices. In the matrix version of the problem, each row and column of an unknown sqrt{n} x sqrt{n} matrix is deleted independently with probability p. Our results contrasts with the best known results for sequence reconstruction where the best known upper bound is exp(O(n^(1/3))).
2) An optimal result for random matrix reconstruction: we show that Theta(log n) traces are necessary and sufficient. This is in contrast to the problem for random sequences where there is a super-logarithmic lower bound and the best known upper bound is exp({O}(log^(1/3) n)).
3) We show that exp(O(k^(1/3) log^(2/3) n)) traces suffice to reconstruct k-sparse strings, providing an improvement over the best known sequence reconstruction results when k = o(n/log^2 n).
4) We show that poly(n) traces suffice if x is k-sparse and we additionally have a "separation" promise, specifically that the indices of 1\u27s in x all differ by Omega(k log n)
Longest Common Subsequence with Gap Constraints
We consider the longest common subsequence problem in the context of
subsequences with gap constraints. In particular, following Day et al. 2022, we
consider the setting when the distance (i. e., the gap) between two consecutive
symbols of the subsequence has to be between a lower and an upper bound (which
may depend on the position of those symbols in the subsequence or on the
symbols bordering the gap) as well as the case where the entire subsequence is
found in a bounded range (defined by a single upper bound), considered by
Kosche et al. 2022. In all these cases, we present effcient algorithms for
determining the length of the longest common constrained subsequence between
two given strings
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