The maximal number of cubic runs in a word
A run is an inclusion maximal occurrence in a word (as a subinterval) of a factor in which the period repeats at least twice. The maximal number of runs in a word of length n has been thoroughly studied, and is known to be between 0.944n and 1.029n. The proofs are very technical. In this paper we in...
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| Veröffentlicht in: | Journal of computer and system sciences 2012-11, Vol.78 (6), p.1828-1836 |
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| container_title | Journal of computer and system sciences |
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| creator | Crochemore, M. Iliopoulos, C.S. Kubica, M. Radoszewski, J. Rytter, W. Waleń, T. |
| description | A run is an inclusion maximal occurrence in a word (as a subinterval) of a factor in which the period repeats at least twice. The maximal number of runs in a word of length n has been thoroughly studied, and is known to be between 0.944n and 1.029n. The proofs are very technical. In this paper we investigate cubic runs, in which the period repeats at least three times. We show the upper bound on their maximal number, cubic-runs(n), in a word of length n: cubic-runs(n) |
| doi_str_mv | 10.1016/j.jcss.2011.12.005 |
| format | Article |
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► We introduce the notion of a cubic run in a word, it is a variant of a run. ► The exact structure of cubic runs in Fibonacci strings is described. ► A lower and an upper bound for the maximal number of cubic runs in a word is given. ► An improved upper bound for binary words is derived.</description><identifier>ISSN: 0022-0000</identifier><identifier>EISSN: 1090-2724</identifier><identifier>DOI: 10.1016/j.jcss.2011.12.005</identifier><language>eng</language><publisher>Elsevier Inc</publisher><subject>Alphabets ; Combinatorics ; Computer Science ; Construction ; Data Structures and Algorithms ; Fibonacci word ; Inclusions ; Linearity ; Lower bounds ; Lyndon word ; Mathematics ; Proving ; Run in a word ; Upper bounds</subject><ispartof>Journal of computer and system sciences, 2012-11, Vol.78 (6), p.1828-1836</ispartof><rights>2011 Elsevier Inc.</rights><rights>Distributed under a Creative Commons Attribution 4.0 International License</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c411t-591506687c19e9aa040152c27c50e78754e6d729cef9de1b3f3c7bf4462603163</citedby><cites>FETCH-LOGICAL-c411t-591506687c19e9aa040152c27c50e78754e6d729cef9de1b3f3c7bf4462603163</cites><orcidid>0000-0002-0067-6401 ; 0000-0003-1087-1419</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://www.sciencedirect.com/science/article/pii/S0022000011001504$$EHTML$$P50$$Gelsevier$$Hfree_for_read</linktohtml><link.rule.ids>230,314,776,780,881,3536,27903,27904,65309</link.rule.ids><backlink>$$Uhttps://hal.science/hal-00836960$$DView record in HAL$$Hfree_for_read</backlink></links><search><creatorcontrib>Crochemore, M.</creatorcontrib><creatorcontrib>Iliopoulos, C.S.</creatorcontrib><creatorcontrib>Kubica, M.</creatorcontrib><creatorcontrib>Radoszewski, J.</creatorcontrib><creatorcontrib>Rytter, W.</creatorcontrib><creatorcontrib>Waleń, T.</creatorcontrib><title>The maximal number of cubic runs in a word</title><title>Journal of computer and system sciences</title><description>A run is an inclusion maximal occurrence in a word (as a subinterval) of a factor in which the period repeats at least twice. The maximal number of runs in a word of length n has been thoroughly studied, and is known to be between 0.944n and 1.029n. The proofs are very technical. In this paper we investigate cubic runs, in which the period repeats at least three times. We show the upper bound on their maximal number, cubic-runs(n), in a word of length n: cubic-runs(n)<0.5n. The proof of linearity of cubic-runs(n) utilizes only simple properties of Lyndon words and is considerably simpler than the corresponding proof for general runs. For binary words, we provide a better upper bound cubic-runs2(n)<0.48n which requires computer-assisted verification of a large number of cases. We also construct an infinite sequence of words over a binary alphabet for which the lower bound is 0.41n.
► We introduce the notion of a cubic run in a word, it is a variant of a run. ► The exact structure of cubic runs in Fibonacci strings is described. ► A lower and an upper bound for the maximal number of cubic runs in a word is given. ► An improved upper bound for binary words is derived.</description><subject>Alphabets</subject><subject>Combinatorics</subject><subject>Computer Science</subject><subject>Construction</subject><subject>Data Structures and Algorithms</subject><subject>Fibonacci word</subject><subject>Inclusions</subject><subject>Linearity</subject><subject>Lower bounds</subject><subject>Lyndon word</subject><subject>Mathematics</subject><subject>Proving</subject><subject>Run in a word</subject><subject>Upper bounds</subject><issn>0022-0000</issn><issn>1090-2724</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2012</creationdate><recordtype>article</recordtype><recordid>eNp9kE9Lw0AQxRdRsFa_gKccVUic2WQ3XfBSiv-g4KWel81mQjekSd1tqn57EyIencvA8HtvZh5j1wgJAsr7OqltCAkHxAR5AiBO2AxBQcxznp2yGQDnMQx1zi5CqGEAhUxn7G6zpWhnvtzONFHb7wryUVdFti-cjXzfhsi1kYk-O19esrPKNIGufvucvT89blYv8frt-XW1XMc2QzzEQqEAKRe5RUXKGMgABbc8twIoX-QiI1nmXFmqVElYpFVq86LKMsklpCjTObudfLem0Xs_XOa_dWecflmu9TgDWKRSSTjiwN5M7N53Hz2Fg965YKlpTEtdHzRiKoVQgvMB5RNqfReCp-rPG0GPIepajyHqMUSNfFgjBtHDJKLh4aMjr4N11FoqnSd70GXn_pP_AL6EdsI</recordid><startdate>20121101</startdate><enddate>20121101</enddate><creator>Crochemore, M.</creator><creator>Iliopoulos, C.S.</creator><creator>Kubica, M.</creator><creator>Radoszewski, J.</creator><creator>Rytter, W.</creator><creator>Waleń, T.</creator><general>Elsevier Inc</general><general>Elsevier</general><scope>6I.</scope><scope>AAFTH</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>8FD</scope><scope>JQ2</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><scope>1XC</scope><scope>VOOES</scope><orcidid>https://orcid.org/0000-0002-0067-6401</orcidid><orcidid>https://orcid.org/0000-0003-1087-1419</orcidid></search><sort><creationdate>20121101</creationdate><title>The maximal number of cubic runs in a word</title><author>Crochemore, M. ; Iliopoulos, C.S. ; Kubica, M. ; Radoszewski, J. ; Rytter, W. ; Waleń, T.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c411t-591506687c19e9aa040152c27c50e78754e6d729cef9de1b3f3c7bf4462603163</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2012</creationdate><topic>Alphabets</topic><topic>Combinatorics</topic><topic>Computer Science</topic><topic>Construction</topic><topic>Data Structures and Algorithms</topic><topic>Fibonacci word</topic><topic>Inclusions</topic><topic>Linearity</topic><topic>Lower bounds</topic><topic>Lyndon word</topic><topic>Mathematics</topic><topic>Proving</topic><topic>Run in a word</topic><topic>Upper bounds</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Crochemore, M.</creatorcontrib><creatorcontrib>Iliopoulos, C.S.</creatorcontrib><creatorcontrib>Kubica, M.</creatorcontrib><creatorcontrib>Radoszewski, J.</creatorcontrib><creatorcontrib>Rytter, W.</creatorcontrib><creatorcontrib>Waleń, T.</creatorcontrib><collection>ScienceDirect Open Access Titles</collection><collection>Elsevier:ScienceDirect:Open Access</collection><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Technology Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><collection>Hyper Article en Ligne (HAL)</collection><collection>Hyper Article en Ligne (HAL) (Open Access)</collection><jtitle>Journal of computer and system sciences</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Crochemore, M.</au><au>Iliopoulos, C.S.</au><au>Kubica, M.</au><au>Radoszewski, J.</au><au>Rytter, W.</au><au>Waleń, T.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>The maximal number of cubic runs in a word</atitle><jtitle>Journal of computer and system sciences</jtitle><date>2012-11-01</date><risdate>2012</risdate><volume>78</volume><issue>6</issue><spage>1828</spage><epage>1836</epage><pages>1828-1836</pages><issn>0022-0000</issn><eissn>1090-2724</eissn><abstract>A run is an inclusion maximal occurrence in a word (as a subinterval) of a factor in which the period repeats at least twice. The maximal number of runs in a word of length n has been thoroughly studied, and is known to be between 0.944n and 1.029n. The proofs are very technical. In this paper we investigate cubic runs, in which the period repeats at least three times. We show the upper bound on their maximal number, cubic-runs(n), in a word of length n: cubic-runs(n)<0.5n. The proof of linearity of cubic-runs(n) utilizes only simple properties of Lyndon words and is considerably simpler than the corresponding proof for general runs. For binary words, we provide a better upper bound cubic-runs2(n)<0.48n which requires computer-assisted verification of a large number of cases. We also construct an infinite sequence of words over a binary alphabet for which the lower bound is 0.41n.
► We introduce the notion of a cubic run in a word, it is a variant of a run. ► The exact structure of cubic runs in Fibonacci strings is described. ► A lower and an upper bound for the maximal number of cubic runs in a word is given. ► An improved upper bound for binary words is derived.</abstract><pub>Elsevier Inc</pub><doi>10.1016/j.jcss.2011.12.005</doi><tpages>9</tpages><orcidid>https://orcid.org/0000-0002-0067-6401</orcidid><orcidid>https://orcid.org/0000-0003-1087-1419</orcidid><oa>free_for_read</oa></addata></record> |
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| source | Elsevier ScienceDirect Journals; Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals |
| subjects | Alphabets Combinatorics Computer Science Construction Data Structures and Algorithms Fibonacci word Inclusions Linearity Lower bounds Lyndon word Mathematics Proving Run in a word Upper bounds |
| title | The maximal number of cubic runs in a word |
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