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4. String inference from longest-common-prefix array

Author

Juha Kärkkäinen, Marcin Piątkowski, Simon J. Puglisi, Chatzigiannakis, Ioannis, Indyk, Piotr, Kuhn, Fabian, Muscholl, Anna, Practical Algorithms and Data Structures on Strings research group / Juha Kärkkäinen, Helsinki Institute for Information Technology, Department of Computer Science, Finnish Centre of Excellence in Algorithmic Data Analysis Research (Algodan), Bioinformatics, Genome-scale Algorithmics research group / Veli Mäkinen, and Algorithmic Bioinformatics

Subjects
String inference, 000 Computer science, knowledge, general works, General Computer Science, LCP array, education, 0102 computer and information sciences, 02 engineering and technology, 113 Computer and information sciences, Quantitative Biology::Genomics, 01 natural sciences, Theoretical Computer Science, 010201 computation theory & mathematics, Computer Science, 0202 electrical engineering, electronic engineering, information engineering, NP-hardness, 020201 artificial intelligence & image processing, Computer Science::Data Structures and Algorithms, Computer Science::Formal Languages and Automata Theory
Abstract

The suffix array, perhaps the most important data structure in modern string processing, is often augmented with the longest common prefix (LCP) array which stores the lengths of the LCPs for lexicographically adjacent suffixes of a string. Together the two arrays are roughly equivalent to the suffix tree with the LCP array representing the tree shape. In order to better understand the combinatorics of LCP arrays, we consider the problem of inferring a string from an LCP array, i.e., determining whether a given array of integers is a valid LCP array, and if it is, reconstructing some string or all strings with that LCP array. There are recent studies of inferring a string from a suffix tree shape but using significantly more information (in the form of suffix links) than is available in the LCP array. We provide two main results. (1) We describe two algorithms for inferring strings from an LCP array when we allow a generalized form of LCP array defined for a multiset of cyclic strings: a linear time algorithm for binary alphabet and a general algorithm with polynomial time complexity for a constant alphabet size. (2) We prove that determining whether a given integer array is a valid LCP array is NP-complete when we require more restricted forms of LCP array defined for a single cyclic or non-cyclic string or a multiset of non-cyclic strings. The result holds whether or not the alphabet is restricted to be binary. In combination, the two results show that the generalized form of LCP array for a multiset of cyclic strings is fundamentally different from the other more restricted forms.

Published
2023

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