Mutual Borders and Overlaps
A word is said to be bordered if it contains a non-empty proper prefix that is also a suffix. We can naturally extend this definition to pairs of non-empty words. A pair of words (u,v) is said to be mutually bordered if there exists a word that is a non-empty proper prefix of u and suffix of v...
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| Veröffentlicht in: | IEEE transactions on information theory 2022-10, Vol.68 (10), p.6888-6893 |
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| description | A word is said to be bordered if it contains a non-empty proper prefix that is also a suffix. We can naturally extend this definition to pairs of non-empty words. A pair of words (u,v) is said to be mutually bordered if there exists a word that is a non-empty proper prefix of u and suffix of v , and there exists a word that is a non-empty proper suffix of u and prefix of v . In other words, (u,v) is mutually bordered if u overlaps v and v overlaps u . We give a recurrence for the number of mutually bordered pairs of words. Furthermore, we show that, asymptotically, there are c\cdot k^{2n} mutually bordered words of length- n over a k -letter alphabet, where c is a constant. Finally, we show that the expected shortest overlap between pairs of words is bounded above by a constant. |
| doi_str_mv | 10.1109/TIT.2022.3167935 |
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We can naturally extend this definition to pairs of non-empty words. A pair of words <inline-formula> <tex-math notation="LaTeX">(u,v) </tex-math></inline-formula> is said to be mutually bordered if there exists a word that is a non-empty proper prefix of <inline-formula> <tex-math notation="LaTeX">u </tex-math></inline-formula> and suffix of <inline-formula> <tex-math notation="LaTeX">v </tex-math></inline-formula>, and there exists a word that is a non-empty proper suffix of <inline-formula> <tex-math notation="LaTeX">u </tex-math></inline-formula> and prefix of <inline-formula> <tex-math notation="LaTeX">v </tex-math></inline-formula>. In other words, <inline-formula> <tex-math notation="LaTeX">(u,v) </tex-math></inline-formula> is mutually bordered if <inline-formula> <tex-math notation="LaTeX">u </tex-math></inline-formula> overlaps <inline-formula> <tex-math notation="LaTeX">v </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">v </tex-math></inline-formula> overlaps <inline-formula> <tex-math notation="LaTeX">u </tex-math></inline-formula>. We give a recurrence for the number of mutually bordered pairs of words. Furthermore, we show that, asymptotically, there are <inline-formula> <tex-math notation="LaTeX">c\cdot k^{2n} </tex-math></inline-formula> mutually bordered words of length-<inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> over a <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula>-letter alphabet, where <inline-formula> <tex-math notation="LaTeX">c </tex-math></inline-formula> is a constant. Finally, we show that the expected shortest overlap between pairs of words is bounded above by a constant.]]></description><identifier>ISSN: 0018-9448</identifier><identifier>EISSN: 1557-9654</identifier><identifier>DOI: 10.1109/TIT.2022.3167935</identifier><identifier>CODEN: IETTAW</identifier><language>eng</language><publisher>New York: IEEE</publisher><subject>bifix-free words ; borders ; Cross-bifix-free codes ; Digital communication ; Distributed databases ; mutually bordered pairs ; non-overlapping codes ; Postal services ; Receivers ; Synchronization ; Terminology</subject><ispartof>IEEE transactions on information theory, 2022-10, Vol.68 (10), p.6888-6893</ispartof><rights>Copyright The Institute of Electrical and Electronics Engineers, Inc. (IEEE) 2022</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c244t-35fc576a44ffa78b0450b8c82f036d503cac13f227347ffff2fbbb419d5312113</cites><orcidid>0000-0001-9707-0803</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://ieeexplore.ieee.org/document/9758763$$EHTML$$P50$$Gieee$$H</linktohtml><link.rule.ids>314,776,780,792,27901,27902,54733</link.rule.ids><linktorsrc>$$Uhttps://ieeexplore.ieee.org/document/9758763$$EView_record_in_IEEE$$FView_record_in_$$GIEEE</linktorsrc></links><search><creatorcontrib>Gabric, Daniel</creatorcontrib><title>Mutual Borders and Overlaps</title><title>IEEE transactions on information theory</title><addtitle>TIT</addtitle><description><![CDATA[A word is said to be bordered if it contains a non-empty proper prefix that is also a suffix. We can naturally extend this definition to pairs of non-empty words. A pair of words <inline-formula> <tex-math notation="LaTeX">(u,v) </tex-math></inline-formula> is said to be mutually bordered if there exists a word that is a non-empty proper prefix of <inline-formula> <tex-math notation="LaTeX">u </tex-math></inline-formula> and suffix of <inline-formula> <tex-math notation="LaTeX">v </tex-math></inline-formula>, and there exists a word that is a non-empty proper suffix of <inline-formula> <tex-math notation="LaTeX">u </tex-math></inline-formula> and prefix of <inline-formula> <tex-math notation="LaTeX">v </tex-math></inline-formula>. In other words, <inline-formula> <tex-math notation="LaTeX">(u,v) </tex-math></inline-formula> is mutually bordered if <inline-formula> <tex-math notation="LaTeX">u </tex-math></inline-formula> overlaps <inline-formula> <tex-math notation="LaTeX">v </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">v </tex-math></inline-formula> overlaps <inline-formula> <tex-math notation="LaTeX">u </tex-math></inline-formula>. We give a recurrence for the number of mutually bordered pairs of words. Furthermore, we show that, asymptotically, there are <inline-formula> <tex-math notation="LaTeX">c\cdot k^{2n} </tex-math></inline-formula> mutually bordered words of length-<inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> over a <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula>-letter alphabet, where <inline-formula> <tex-math notation="LaTeX">c </tex-math></inline-formula> is a constant. Finally, we show that the expected shortest overlap between pairs of words is bounded above by a constant.]]></description><subject>bifix-free words</subject><subject>borders</subject><subject>Cross-bifix-free codes</subject><subject>Digital communication</subject><subject>Distributed databases</subject><subject>mutually bordered pairs</subject><subject>non-overlapping codes</subject><subject>Postal services</subject><subject>Receivers</subject><subject>Synchronization</subject><subject>Terminology</subject><issn>0018-9448</issn><issn>1557-9654</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2022</creationdate><recordtype>article</recordtype><sourceid>RIE</sourceid><recordid>eNo9kE1LxDAQhoMoWFfvwl4KnlszmUnTHHXxY2FlL_Uc0jaBXeq2Jq3gv7fLLs7lZeB5Z-Bh7B54DsD1Y7WucsGFyBEKpVFesASkVJkuJF2yhHMoM01UXrObGPfzShJEwpYf0zjZLn3uQ-tCTO2hTbc_LnR2iLfsytsuurtzLtjn60u1es8227f16mmTNYJozFD6RqrCEnlvVVlzkrwum1J4jkUrOTa2AfRCKCTl5xG-rmsC3UoEAYAL9nC6O4T-e3JxNPt-Cof5pREKSEsCxJniJ6oJfYzBeTOE3ZcNvwa4OSowswJzVGDOCubK8lTZOef-ca1kqQrEP1kPVQE</recordid><startdate>20221001</startdate><enddate>20221001</enddate><creator>Gabric, Daniel</creator><general>IEEE</general><general>The Institute of Electrical and Electronics Engineers, Inc. (IEEE)</general><scope>97E</scope><scope>RIA</scope><scope>RIE</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>7SP</scope><scope>8FD</scope><scope>JQ2</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><orcidid>https://orcid.org/0000-0001-9707-0803</orcidid></search><sort><creationdate>20221001</creationdate><title>Mutual Borders and Overlaps</title><author>Gabric, Daniel</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c244t-35fc576a44ffa78b0450b8c82f036d503cac13f227347ffff2fbbb419d5312113</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2022</creationdate><topic>bifix-free words</topic><topic>borders</topic><topic>Cross-bifix-free codes</topic><topic>Digital communication</topic><topic>Distributed databases</topic><topic>mutually bordered pairs</topic><topic>non-overlapping codes</topic><topic>Postal services</topic><topic>Receivers</topic><topic>Synchronization</topic><topic>Terminology</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Gabric, Daniel</creatorcontrib><collection>IEEE All-Society Periodicals Package (ASPP) 2005-present</collection><collection>IEEE All-Society Periodicals Package (ASPP) 1998-Present</collection><collection>IEEE Electronic Library (IEL)</collection><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Electronics & Communications 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><jtitle>IEEE transactions on information theory</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Gabric, Daniel</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Mutual Borders and Overlaps</atitle><jtitle>IEEE transactions on information theory</jtitle><stitle>TIT</stitle><date>2022-10-01</date><risdate>2022</risdate><volume>68</volume><issue>10</issue><spage>6888</spage><epage>6893</epage><pages>6888-6893</pages><issn>0018-9448</issn><eissn>1557-9654</eissn><coden>IETTAW</coden><abstract><![CDATA[A word is said to be bordered if it contains a non-empty proper prefix that is also a suffix. We can naturally extend this definition to pairs of non-empty words. A pair of words <inline-formula> <tex-math notation="LaTeX">(u,v) </tex-math></inline-formula> is said to be mutually bordered if there exists a word that is a non-empty proper prefix of <inline-formula> <tex-math notation="LaTeX">u </tex-math></inline-formula> and suffix of <inline-formula> <tex-math notation="LaTeX">v </tex-math></inline-formula>, and there exists a word that is a non-empty proper suffix of <inline-formula> <tex-math notation="LaTeX">u </tex-math></inline-formula> and prefix of <inline-formula> <tex-math notation="LaTeX">v </tex-math></inline-formula>. In other words, <inline-formula> <tex-math notation="LaTeX">(u,v) </tex-math></inline-formula> is mutually bordered if <inline-formula> <tex-math notation="LaTeX">u </tex-math></inline-formula> overlaps <inline-formula> <tex-math notation="LaTeX">v </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">v </tex-math></inline-formula> overlaps <inline-formula> <tex-math notation="LaTeX">u </tex-math></inline-formula>. We give a recurrence for the number of mutually bordered pairs of words. Furthermore, we show that, asymptotically, there are <inline-formula> <tex-math notation="LaTeX">c\cdot k^{2n} </tex-math></inline-formula> mutually bordered words of length-<inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> over a <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula>-letter alphabet, where <inline-formula> <tex-math notation="LaTeX">c </tex-math></inline-formula> is a constant. Finally, we show that the expected shortest overlap between pairs of words is bounded above by a constant.]]></abstract><cop>New York</cop><pub>IEEE</pub><doi>10.1109/TIT.2022.3167935</doi><tpages>6</tpages><orcidid>https://orcid.org/0000-0001-9707-0803</orcidid></addata></record> |
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| subjects | bifix-free words borders Cross-bifix-free codes Digital communication Distributed databases mutually bordered pairs non-overlapping codes Postal services Receivers Synchronization Terminology |
| title | Mutual Borders and Overlaps |
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