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Range LCP. (English) Zbl 1410.68414
Summary: In this paper, we define the Range LCP problem as follows. Preprocess a string \(S\), of length \(n\), to enable efficient solutions of the following query: Given \([i, j]\), \(0<i\leqslant j\leqslant n\), compute \(\max_{\ell,k \in [i..j]}\mathrm{LCP}(S_\ell,S_k)\), where \(\mathrm{LCP}(S_\ell, S_k)\) is the length of the longest common prefix of the suffixes of \(S\) starting at locations \(\ell\) and \(k\). This is a natural generalization of the classical LCP problem. We provide algorithms with the following complexities:
1.
Preprocessing Time: \(O(|S|)\), Space: \(O(|S|)\), Query Time: \(O(|j-i|\log\log n)\).
2.
Preprocessing Time: none, Space: \(O(|j-i|\log |j-i|)\), Query Time: \(O(|j-i|\log |j-i|)\). However, the query just gives the pairs with the longest LCP, not the LCP itself.
3.
Preprocessing Time: \(O(|S| \log^2 |S|)\), Space: \(O(|S| \log^{1+\varepsilon}|S|)\) for arbitrary small constant \(\varepsilon\), Query Time: \(O(\log\log |S|)\).

MSC:
68W32 Algorithms on strings
68P05 Data structures
68W40 Analysis of algorithms
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[1] Apostolico, A.; Preparata, F. P., Optimal off-line detection of repetitions in a string, Theor. Comput. Sci., 22, 297-315, (1983) · Zbl 0497.68052
[2] Bender, M. A.; Farach-Colton, M., The LCA problem revisited, (LATIN, (2000)), 88-94 · Zbl 0959.68133
[3] Berkman, O.; Breslauer, D.; Galil, Z.; Schieber, B.; Vishkin, U., Highly parallelizable problems, (Proc. 21st ACM Symposium on Theory of Computation, (1989)), 309-319
[4] Chan, T. M.; Larsen, K. G.; Pǎtraşcu, M., Orthogonal range searching on the ram, revisited, (Proc. 27th ACM Symposium on Computational Geometry, (SoCG), (2011)), 1-10 · Zbl 1283.68139
[5] Cormode, G.; Muthukrishnan, S., Substring compression problems, (Proc. 16th Annual ACM-SIAM Symposium on Discrete Algorithms, (SODA), (2005)), 321-330 · Zbl 1297.68278
[6] Farach, M., Optimal suffix tree construction with large alphabets, (Proc. 38th IEEE Symposium on Foundations of Computer Science, (1997)), 137-143
[7] Harel, D.; Tarjan, R. E., Fast algorithms for finding nearest common ancestor, J. Comput. Syst. Sci., 13, 338-355, (1984) · Zbl 0535.68022
[8] Iacono, J.; Özkan, O., Mergeable dictionaries, (Proc. 37th International Colloquium on Automata, Languages and Programming (ICALP), vol. 1, Lect. Notes Comput. Sci., vol. 6198, (2010), Springer), 164-175 · Zbl 1287.68030
[9] Ilie, L.; Tinta, L., Practical algorithms for the longest common extension problem, (Proc. 16th International Symposium on String Processing and Information Retrieval, (SPIRE), Lect. Notes Comput. Sci., vol. 5721, (2009), Springer), 302-309
[10] Kärkkäinen, J.; Sanders, P., Simple linear work suffix array construction, (Proc. 30th International Colloquium on Automata, Languages and Programming, (ICALP), Lect. Notes Comput. Sci., vol. 2719, (2003)), 943-955 · Zbl 1039.68042
[11] Kasai, T.; Lee, G.; Arimura, H.; Arikawa, S.; Park, K., Linear-time longest-common-prefix computation in suffix arrays and its applications, (Proc. 12th Symposium on Combinatorial Pattern Matching, (CPM), (2001)), 181-192 · Zbl 0990.68639
[12] Keller, O.; Kopelowitz, T.; Landau, S.; Lewenstein, M., Generalized substring compression, (Proc. 20th Annual Symposium on Combinatorial Pattern Matching (CPM), (2009), Springer), 26-38
[13] Landau, G. M.; Vishkin, U., Efficient string matching in the presence of errors, (Proc. 26th IEEE FOCS, (1985)), 126-136
[14] Landau, G. M.; Vishkin, U., Fast parallel and serial approximate string matching, J. Algorithms, 10, 2, 157-169, (1989) · Zbl 0685.68033
[15] Lempel, A.; Ziv, J., On the complexity of finite sequences, IEEE Trans. Inf. Theory, 22, 75-81, (1976) · Zbl 0337.94013
[16] McCreight, E. M., A space-economical suffix tree construction algorithm, J. ACM, 23, 262-272, (1976) · Zbl 0329.68042
[17] Ukkonen, E., On-line construction of suffix trees, Algorithmica, 14, 249-260, (1995) · Zbl 0831.68027
[18] van Emde Boas, P.; Kaas, R.; Zijlstra, E., Design and implementation of an efficient priority queue, Math. Syst. Theory, 10, 99-127, (1977) · Zbl 0363.60104
[19] Weiner, P., Linear pattern matching algorithm, (Proc. 14th IEEE Symposium on Switching and Automata Theory, (1973)), 1-11
[20] Yuan, H.; Atallah, M. J., Data structures for range minimum queries in multidimensional arrays, (Proc. 21st ACM-SIAM Symposium on Discrete Algorithms, (SODA), (2010)), 150-160 · Zbl 1288.68055
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