Closest periodic vectors in Lp spaces
The problem of finding the period of a vector V is central to many applications. Let V′ be a periodic vector closest to V under some metric. We seek this V′, or more precisely we seek the smallest period that generates V′. In this paper we consider the problem of finding the closest periodic vector...
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| Vydáno v: | Theoretical computer science Ročník 533; s. 26 - 36 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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Elsevier B.V
08.05.2014
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| ISSN: | 0304-3975, 1879-2294 |
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| Abstract | The problem of finding the period of a vector V is central to many applications. Let V′ be a periodic vector closest to V under some metric. We seek this V′, or more precisely we seek the smallest period that generates V′. In this paper we consider the problem of finding the closest periodic vector in Lp spaces. The measures of “closeness” that we consider are the metrics in the different Lp spaces. Specifically, we consider the L1,L2 and L∞ metrics. In particular, for a given n-dimensional vector V, we develop O(n2) time algorithms (a different algorithm for each metric) that construct the smallest period that defines such a periodic n-dimensional vector V′. We call that vector the closest periodic vector of V under the appropriate metric. We also show (three) O˜(n) time constant approximation algorithms for the period of the approximate closest periodic vector. |
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| AbstractList | The problem of finding the period of a vector V is central to many applications. Let V′ be a periodic vector closest to V under some metric. We seek this V′, or more precisely we seek the smallest period that generates V′. In this paper we consider the problem of finding the closest periodic vector in Lp spaces. The measures of “closeness” that we consider are the metrics in the different Lp spaces. Specifically, we consider the L1,L2 and L∞ metrics. In particular, for a given n-dimensional vector V, we develop O(n2) time algorithms (a different algorithm for each metric) that construct the smallest period that defines such a periodic n-dimensional vector V′. We call that vector the closest periodic vector of V under the appropriate metric. We also show (three) O˜(n) time constant approximation algorithms for the period of the approximate closest periodic vector. |
| Author | Amir, Amihood Eisenberg, Estrella Levy, Avivit Lewenstein, Noa |
| Author_xml | – sequence: 1 givenname: Amihood surname: Amir fullname: Amir, Amihood email: amir@cs.biu.ac.il organization: Department of Computer Science, Bar-Ilan University, Ramat-Gan 52900, Israel – sequence: 2 givenname: Estrella surname: Eisenberg fullname: Eisenberg, Estrella organization: Department of Computer Science, Bar-Ilan University, Ramat-Gan 52900, Israel – sequence: 3 givenname: Avivit surname: Levy fullname: Levy, Avivit email: avivitlevy@shenkar.ac.il organization: Department of Software Engineering, Shenkar College, 12 Anna Frank, Ramat-Gan, Israel – sequence: 4 givenname: Noa surname: Lewenstein fullname: Lewenstein, Noa email: noa.lewenstein@gmail.com organization: Netanya College, Netanya, Israel |
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| Copyright | 2014 Elsevier B.V. |
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| Keywords | String algorithms Closest vector Approximate periodicity |
| Language | English |
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| References | Amir, Benson, Farach (br0030) 1993 Cole, Crochemore, Galil, Gąsieniec, Harihan, Muthukrishnan, Park, Rytter (br0080) 1993 Amir, Benson (br0020) 1998; 27 Régnier, Rostami (br0190) 1993; vol. 684 Amir, Eisenberg, Levy, Porat, Shapira (br0060) 2010 Gfeller (br0140) 2011 Apostolico, Giancarlo (br0070) 2008; 156 Crochemore (br0110) 1981; 12 Landau, Schmidt, Sokol (br0150) 2001; 8 Ma, Hellerstein (br0180) 2001 Stout (br0200) 2008; 53 Amir, Benson, Farach (br0040) 1998; 144 Amir, Eisenberg, Levy (br0050) 2010 Lothaire (br0170) 1983 Lipsky, Porat (br0160) 2011; 60 Cormen, Leiserson, Rivest (br0100) 1992 Galil, Park (br0130) 1996; 25 br0010 Cole, Crochemore, Galil, Gasieniec, Hariharan, Muthukrishnan, Park, Ramesh, Rytter (br0090) 1993 Galil (br0120) 1984; vol. 67 |
| References_xml | – start-page: 205 year: 2001 end-page: 214 ident: br0180 article-title: Mining partially periodic event patterns with unknown periods publication-title: The 17th International Conference on Data Engineering (ICDE) – volume: 156 start-page: 1389 year: 2008 end-page: 1398 ident: br0070 article-title: Periodicity and repetitions in parameterized strings publication-title: Discrete Appl. Math. – year: 1983 ident: br0170 article-title: Combinatorics on Words – volume: 60 start-page: 335 year: 2011 end-page: 348 ident: br0160 article-title: Approximated pattern matching with the publication-title: Algorithmica – volume: 12 start-page: 244 year: 1981 end-page: 250 ident: br0110 article-title: An optimal algorithm for computing the repetitions in a word publication-title: Inform. Process. Lett. – year: 1992 ident: br0100 article-title: Introduction to Algorithms – start-page: 43 year: 2010 end-page: 54 ident: br0060 article-title: Cycle detection and correction publication-title: The 37th International Colloquium on Automata, Languages and Programming (ICALP) – volume: 27 start-page: 90 year: 1998 end-page: 106 ident: br0020 article-title: Two-dimensional periodicity and its application publication-title: SIAM J. Comput. – start-page: 25 year: 2010 end-page: 36 ident: br0050 article-title: Approximate periodicity publication-title: The 21st International Symposium on Algorithms and Computation (ISAAC) – ident: br0010 – start-page: 248 year: 1993 end-page: 258 ident: br0090 article-title: Optimally fast parallel algorithms for preprocessing and pattern matching in one and two dimensions publication-title: Proc. 34th Annual IEEE FOCS – start-page: 79 year: 1993 end-page: 85 ident: br0030 article-title: Optimal parallel two dimensional pattern matching publication-title: Proc. of the 5th ACM Symp. on Parallel Algorithms and Architectures – volume: 25 start-page: 907 year: 1996 end-page: 935 ident: br0130 article-title: Alphabet-independent two-dimensional witness computation publication-title: SIAM J. Comput. – start-page: 463 year: 2011 end-page: 474 ident: br0140 article-title: Finding longest approximate periodic patterns publication-title: WADS – volume: vol. 67 start-page: 144 year: 1984 end-page: 157 ident: br0120 article-title: Optimal parallel algorithms for string matching publication-title: Proc. 16th ACM Symposium on Theory of Computing – volume: 8 start-page: 1 year: 2001 end-page: 18 ident: br0150 article-title: An algorithm for approximate tandem repeats publication-title: J. Comput. Biol. – volume: vol. 684 start-page: 215 year: 1993 end-page: 227 ident: br0190 article-title: A unifying look at publication-title: Proc. 4th Symp. on Combinatorial Pattern Matching – volume: 53 start-page: 289 year: 2008 end-page: 297 ident: br0200 article-title: Unimodal regression via prefix isotonic regression publication-title: Comput. Statist. Data Anal. – volume: 144 start-page: 1 year: 1998 end-page: 17 ident: br0040 article-title: Optimal parallel two dimensional text searching on a crew pram publication-title: Inform. and Comput. – start-page: 248 year: 1993 end-page: 258 ident: br0080 article-title: Optimally fast parallel algorithms for preprocessing and pattern matching in one and two dimensions publication-title: Proc. 34th IEEE FOCS |
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| SubjectTerms | Approximate periodicity Closest vector String algorithms |
| Title | Closest periodic vectors in Lp spaces |
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