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Time-Space Trade-Offs for Longest Common Extensions [article]

Philip Bille, Inge Li Goertz, Benjamin Sach, Hjalte Wedel Vildhøj
2013 arXiv   pre-print
We study the time-space trade-offs for the problem, that is, the space used for the data structure vs. the worst-case time for answering an LCE query. Let n be the length of T.  ...  We revisit the longest common extension (LCE) problem, that is, preprocess a string T into a compact data structure that supports fast LCE queries.  ...  The longest common extension problem (LCE problem) is to preprocess T into a compact data structure supporting fast longest common extension queries.  ... 
arXiv:1211.0270v2 fatcat:7ovn7espzfg7nfeg3f2xtemulu

Time-Space Trade-Offs for Longest Common Extensions [chapter]

Philip Bille, Inge Li Gørtz, Benjamin Sach, Hjalte Wedel Vildhøj
2012 Lecture Notes in Computer Science  
We study the time-space trade-offs for the problem, that is, the space used for the data structure vs. the worst-case time for answering an LCE query. Let n be the length of T .  ...  We revisit the longest common extension (LCE) problem, that is, preprocess a string T into a compact data structure that supports fast LCE queries.  ...  Acknowledgements We thank the anonymous referes for their valuable comments, and in particular for pointing out an issue with the preprocessing space of the randomised solution.  ... 
doi:10.1007/978-3-642-31265-6_24 fatcat:puexhxxzjnfjpmhda3qtbhsm6m

Time–space trade-offs for longest common extensions

Philip Bille, Inge Li Gørtz, Benjamin Sach, Hjalte Wedel Vildhøj
2014 Journal of Discrete Algorithms  
We study the time-space trade-offs for the problem, that is, the space used for the data structure vs. the worst-case time for answering an LCE query. Let n be the length of T .  ...  We revisit the longest common extension (LCE) problem, that is, preprocess a string T into a compact data structure that supports fast LCE queries.  ...  Acknowledgements We thank the anonymous referes for their valuable comments, and in particular for pointing out an issue with the preprocessing space of the randomised solution.  ... 
doi:10.1016/j.jda.2013.06.003 fatcat:tyahwybryvb7rlbgtclx6bgcwq

Tight Lower Bounds for the Longest Common Extension Problem [article]

Dmitry Kosolobov
2017 arXiv   pre-print
The longest common extension problem is to preprocess a given string of length n into a data structure that uses S(n) bits on top of the input and answers in T(n) time the queries LCE(i,j) computing the  ...  It is known that this trade-off is tight.  ...  The author wishes to acknowledge anonymous referees for detailed comments that helped to greatly improve the paper.  ... 
arXiv:1611.02891v5 fatcat:cut32cw3rjcrtdrtxjcu2zc5si

Longest Common Extensions in Sublinear Space [chapter]

Philip Bille, Inge Li Gørtz, Mathias Bæk Tejs Knudsen, Moshe Lewenstein, Hjalte Wedel Vildhøj
2015 Lecture Notes in Computer Science  
In this paper we show that for any trade-off parameter 1 ≤ τ ≤ n, the problem can be solved in O( n τ ) space and O(τ ) query time.  ...  The longest common extension problem (LCE problem) is to construct a data structure for an input string T of length n that supports LCE(i, j) queries.  ...  The longest common extension problem (LCE problem) is to preprocess T into a compact data structure supporting fast longest common extension queries.  ... 
doi:10.1007/978-3-319-19929-0_6 fatcat:jedgbw4fdrcptlx4cwuflpaabu

Longest Common Extensions in Sublinear Space [article]

Philip Bille, Inge Li Gørtz, Mathias Bæk Tejs Knudsen, Moshe Lewenstein, Hjalte Wedel Vildhøj
2015 arXiv   pre-print
In this paper we show that for any trade-off parameter 1 ≤τ≤ n, the problem can be solved in O(n/τ) space and O(τ) query time.  ...  The longest common extension problem (LCE problem) is to construct a data structure for an input string T of length n that supports LCE(i,j) queries.  ...  The longest common extension problem (LCE problem) is to preprocess T into a compact data structure supporting fast longest common extension queries.  ... 
arXiv:1504.02671v1 fatcat:virzw24iijaivpy5oyed3w3bma

Substring Complexity in Sublinear Space [article]

Giulia Bernardini and Gabriele Fici and Paweł Gawrychowski and Solon P. Pissis
2020 arXiv   pre-print
We present the following results: an O(n^3/b^2)-time and O(b)-space algorithm to compute δ, for any b∈[1,n]; and an Õ(n^2/b)-time and O(b)-space algorithm to compute δ, for any b∈[n^2/3,n].  ...  In particular, δ≤γ always holds and δ can be computed in O(n) time using Ω(n) working space.  ...  Given a string T of length n, we can compute δ = sup{S T (k)/k, k ≥ 1} inÕ(n 2 /b) time using O(b) space, for any b ∈ [n 2/3 , n].  ... 
arXiv:2007.08357v1 fatcat:r54pfdn6tvfztcpg4h7lvgp4qm

Page 1171 of Mathematical Reviews Vol. , Issue 85c [page]

1985 Mathematical Reviews  
Ja’Ja’, Joseph (1-PAS) 85c:68026 Time-space trade-offs for some algebraic problems. J. Assoc. Comput. Mach. 30 (1983), no. 3, 657-667.  ...  For an application to pattern recognition, a discussion is presented on extensions to weighted longest common subsequences and to the longest common subma- trix.” Vodop’yanov, V. K.; Zavyalov, V.  ... 

Sampled Longest Common Prefix Array [chapter]

Jouni Sirén
2010 Lecture Notes in Computer Science  
When augmented with the longest common prefix (LCP) array and some other structures, the suffix array can solve many string processing problems in optimal time and space.  ...  We show that when used with a compressed suffix array (CSA), the sampled LCP array often offers better time/space trade-offs than the existing alternatives.  ...  While the suffix array and the LCP array can be compressed to a space relative to the number of of equal letter runs in BWT, no such representation is known for suffix tree topology.  ... 
doi:10.1007/978-3-642-13509-5_21 fatcat:ahsnbuzbyrgz3fydzi3iqewkrq

Longest common extensions in trees

Philip Bille, Paweł Gawrychowski, Inge Li Gørtz, Gad M. Landau, Oren Weimann
2016 Theoretical Computer Science  
For LCETT queries, we present a time-space trade-off, that given any parameter τ , 1 ≤ τ ≤ n, leads to an O(nτ ) space and O(n/τ ) query-time solution (all of these bounds hold on a standard unit-cost  ...  The longest common extension (LCE) of two indices in a string is the length of the longest identical substrings starting at these two indices.  ...  The Time-Space Trade-Off We now give a time-space trade-off for the LCE TT problem as stated by the following theorem.  ... 
doi:10.1016/j.tcs.2015.08.009 fatcat:agclladhdnbobatxasj27rkuwu

Longest Common Extensions in Trees [article]

Philip Bille, Pawel Gawrychowski, Inge Li Goertz, Gad M. Landau, and Oren Weimann
2015 arXiv   pre-print
For queries, we present a time-space trade-off, that given any parameter τ, 1 ≤τ≤ n, leads to an O(nτ) space and O(n/τ) query-time solution.  ...  The longest common extension (LCE) of two indices in a string is the length of the longest identical substrings starting at these two indices.  ...  The Time-Space Trade-Off We now give a time-space trade-off for the LCE TT problem as stated by the following theorem.  ... 
arXiv:1412.1254v3 fatcat:ehijuinpqjh7hca5ljhomihgjq

Longest Common Extensions in Trees [chapter]

Philip Bille, Paweł Gawrychowski, Inge Li Gørtz, Gad M. Landau, Oren Weimann
2015 Lecture Notes in Computer Science  
For LCETT queries, we present a time-space trade-off, that given any parameter τ , 1 ≤ τ ≤ n, leads to an O(nτ ) space and O(n/τ ) query-time solution.  ...  The longest common extension (LCE) of two indices in a string is the length of the longest identical substrings starting at these two indices.  ...  We complement this result with a time-space trade-off that achieves O(nτ ) space and O(n/τ ) query time for any parameter 1 ≤ τ ≤ n.  ... 
doi:10.1007/978-3-319-19929-0_5 fatcat:7dhqd3ppajaufe4g3zaftoakvi

Sublinear Space Algorithms for the Longest Common Substring Problem [chapter]

Tomasz Kociumaka, Tatiana Starikovskaya, Hjalte Wedel Vildhøj
2014 Lecture Notes in Computer Science  
In this paper we show that for any trade-off parameter 1 ≤ τ ≤ n, the LCS problem can be solved in O(τ ) space and O(n 2 /τ ) time, thus providing the first smooth deterministic timespace trade-off from  ...  This problem is known as the longest common substring (LCS) problem and has a classic O(n) space and O(n) time solution (Weiner [FOCS'73], Hui [CPM'92]).  ...  Below we show a time-space trade-off lower bound for element bidistinctness. Lemma 9. If |D| ≥ 2n 2 , at least a fraction 1/e of inputs belong to EB −1 (1). Proof.  ... 
doi:10.1007/978-3-662-44777-2_50 fatcat:lsgrntujabfenn7zv36krheuma

Sublinear Space Algorithms for the Longest Common Substring Problem [article]

Tomasz Kociumaka, Tatiana Starikovskaya, Hjalte Wedel Vildhøj
2014 arXiv   pre-print
In this paper we show that for any trade-off parameter 1 ≤τ≤ n, the LCS problem can be solved in O(τ) space and O(n^2/τ) time, thus providing the first smooth deterministic time-space trade-off from constant  ...  This problem is known as the longest common substring (LCS) problem and has a classic O(n) space and O(n) time solution (Weiner [FOCS'73], Hui [CPM'92]).  ...  Below we show a time-space trade-off lower bound for element bidistinctness. Lemma 9. If |D| ≥ 2n 2 , at least a fraction 1/e of inputs belong to EB −1 (1). Proof.  ... 
arXiv:1407.0522v1 fatcat:reuwyixidjb5dchgvrsfgrzvzy

Fast Longest Common Extensions in Small Space [article]

Alberto Policriti, Nicola Prezza
2016 arXiv   pre-print
In this paper we address the longest common extension (LCE) problem: to compute the length ℓ of the longest common prefix between any two suffixes of T∈Σ^n with Σ = {0, ...σ-1} .  ...  Our results improve the state of the art in space usage, query times, and preprocessing times and are extremely practical: we present a C++ implementation that is very fast and space-efficient in practice  ...  The longest common extension (LCE) problem asks to build a data structure on T supporting fast LCE queries: given 0 ≤ i, j < n, return the length ℓ = LCE(i, j) of the longest common prefix of T [i, . .  ... 
arXiv:1607.06660v1 fatcat:2bcauasygrhkrdsy253fzznjmm
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