{"id":"https://openalex.org/W3155567177","doi":"https://doi.org/10.3390/a14040126","title":"Branching Densities of Cube-Free and Square-Free Words","display_name":"Branching Densities of Cube-Free and Square-Free Words","publication_year":2021,"publication_date":"2021-04-20","ids":{"openalex":"https://openalex.org/W3155567177","doi":"https://doi.org/10.3390/a14040126","mag":"3155567177"},"language":"en","primary_location":{"id":"doi:10.3390/a14040126","is_oa":true,"landing_page_url":"https://doi.org/10.3390/a14040126","pdf_url":"https://www.mdpi.com/1999-4893/14/4/126/pdf?version=1618927534","source":{"id":"https://openalex.org/S190629608","display_name":"Algorithms","issn_l":"1999-4893","issn":["1999-4893"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Algorithms","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1999-4893/14/4/126/pdf?version=1618927534","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5007984577","display_name":"Elena A. Petrova","orcid":"https://orcid.org/0000-0002-2298-1431"},"institutions":[{"id":"https://openalex.org/I130823665","display_name":"Ural Federal University","ror":"https://ror.org/00hs7dr46","country_code":"RU","type":"education","lineage":["https://openalex.org/I130823665"]}],"countries":["RU"],"is_corresponding":false,"raw_author_name":"Elena A. Petrova","raw_affiliation_strings":["Department of Algebra and Fundamental Informatics, Ural Federal University, 620075 Yekaterinburg, Russia"],"affiliations":[{"raw_affiliation_string":"Department of Algebra and Fundamental Informatics, Ural Federal University, 620075 Yekaterinburg, Russia","institution_ids":["https://openalex.org/I130823665"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5063068146","display_name":"Arseny M. Shur","orcid":"https://orcid.org/0000-0002-7812-3399"},"institutions":[{"id":"https://openalex.org/I130823665","display_name":"Ural Federal University","ror":"https://ror.org/00hs7dr46","country_code":"RU","type":"education","lineage":["https://openalex.org/I130823665"]}],"countries":["RU"],"is_corresponding":true,"raw_author_name":"Arseny M. Shur","raw_affiliation_strings":["Department of Algebra and Fundamental Informatics, Ural Federal University, 620075 Yekaterinburg, Russia"],"affiliations":[{"raw_affiliation_string":"Department of Algebra and Fundamental Informatics, Ural Federal University, 620075 Yekaterinburg, Russia","institution_ids":["https://openalex.org/I130823665"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5063068146"],"corresponding_institution_ids":["https://openalex.org/I130823665"],"apc_list":{"value":1400,"currency":"CHF","value_usd":1515},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1515},"fwci":0.1507,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.49107768,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":93},"biblio":{"volume":"14","issue":"4","first_page":"126","last_page":"126"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11567","display_name":"semigroups and automata theory","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11567","display_name":"semigroups and automata theory","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11269","display_name":"Algorithms and Data Compression","score":0.9984999895095825,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12029","display_name":"DNA and Biological Computing","score":0.9979000091552734,"subfield":{"id":"https://openalex.org/subfields/1312","display_name":"Molecular Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/binary-tree","display_name":"Binary tree","score":0.6452738046646118},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6351267099380493},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6098577380180359},{"id":"https://openalex.org/keywords/branching","display_name":"Branching (polymer chemistry)","score":0.6090285778045654},{"id":"https://openalex.org/keywords/ternary-operation","display_name":"Ternary operation","score":0.5579730272293091},{"id":"https://openalex.org/keywords/square","display_name":"Square (algebra)","score":0.5460793972015381},{"id":"https://openalex.org/keywords/cube","display_name":"Cube (algebra)","score":0.530632734298706},{"id":"https://openalex.org/keywords/binary-number","display_name":"Binary number","score":0.5297008156776428},{"id":"https://openalex.org/keywords/ternary-search-tree","display_name":"Ternary search tree","score":0.5150041580200195},{"id":"https://openalex.org/keywords/prefix","display_name":"Prefix","score":0.48745131492614746},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4512813985347748},{"id":"https://openalex.org/keywords/binary-search-tree","display_name":"Binary search tree","score":0.44109833240509033},{"id":"https://openalex.org/keywords/tree","display_name":"Tree (set theory)","score":0.41856417059898376},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.23363003134727478},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.19616550207138062},{"id":"https://openalex.org/keywords/tree-structure","display_name":"Tree structure","score":0.1581535041332245},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.12203291058540344},{"id":"https://openalex.org/keywords/interval-tree","display_name":"Interval tree","score":0.0867287814617157},{"id":"https://openalex.org/keywords/chemistry","display_name":"Chemistry","score":0.08398514986038208}],"concepts":[{"id":"https://openalex.org/C197855036","wikidata":"https://www.wikidata.org/wiki/Q380172","display_name":"Binary tree","level":2,"score":0.6452738046646118},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6351267099380493},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6098577380180359},{"id":"https://openalex.org/C206175624","wikidata":"https://www.wikidata.org/wiki/Q595731","display_name":"Branching (polymer chemistry)","level":2,"score":0.6090285778045654},{"id":"https://openalex.org/C64452783","wikidata":"https://www.wikidata.org/wiki/Q1524945","display_name":"Ternary operation","level":2,"score":0.5579730272293091},{"id":"https://openalex.org/C135692309","wikidata":"https://www.wikidata.org/wiki/Q111124","display_name":"Square (algebra)","level":2,"score":0.5460793972015381},{"id":"https://openalex.org/C53051483","wikidata":"https://www.wikidata.org/wiki/Q861555","display_name":"Cube (algebra)","level":2,"score":0.530632734298706},{"id":"https://openalex.org/C48372109","wikidata":"https://www.wikidata.org/wiki/Q3913","display_name":"Binary number","level":2,"score":0.5297008156776428},{"id":"https://openalex.org/C136519935","wikidata":"https://www.wikidata.org/wiki/Q2859802","display_name":"Ternary search tree","level":5,"score":0.5150041580200195},{"id":"https://openalex.org/C141603448","wikidata":"https://www.wikidata.org/wiki/Q134830","display_name":"Prefix","level":2,"score":0.48745131492614746},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4512813985347748},{"id":"https://openalex.org/C91154448","wikidata":"https://www.wikidata.org/wiki/Q623818","display_name":"Binary search tree","level":3,"score":0.44109833240509033},{"id":"https://openalex.org/C113174947","wikidata":"https://www.wikidata.org/wiki/Q2859736","display_name":"Tree (set theory)","level":2,"score":0.41856417059898376},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.23363003134727478},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.19616550207138062},{"id":"https://openalex.org/C163797641","wikidata":"https://www.wikidata.org/wiki/Q2067937","display_name":"Tree structure","level":3,"score":0.1581535041332245},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.12203291058540344},{"id":"https://openalex.org/C84927040","wikidata":"https://www.wikidata.org/wiki/Q6057306","display_name":"Interval tree","level":4,"score":0.0867287814617157},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.08398514986038208},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C178790620","wikidata":"https://www.wikidata.org/wiki/Q11351","display_name":"Organic chemistry","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.3390/a14040126","is_oa":true,"landing_page_url":"https://doi.org/10.3390/a14040126","pdf_url":"https://www.mdpi.com/1999-4893/14/4/126/pdf?version=1618927534","source":{"id":"https://openalex.org/S190629608","display_name":"Algorithms","issn_l":"1999-4893","issn":["1999-4893"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Algorithms","raw_type":"journal-article"},{"id":"pmh:oai:elar.urfu.ru:10995/102933","is_oa":true,"landing_page_url":"https://www.scopus.com/inward/record.uri?eid=2-s2.0-85105119295&doi=10.3390%2fa14040126&partnerID=40&md5=3dcf4125418835cf85bf3ac537dfebc9","pdf_url":"https://www.mdpi.com/1999-4893/14/4/126/pdf","source":{"id":"https://openalex.org/S4306401522","display_name":"Electronic scientific archive of UrFU (Ural Federal University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I130823665","host_organization_name":"Ural Federal University","host_organization_lineage":["https://openalex.org/I130823665"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Algorithms","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:doaj.org/article:7a0de29ce7294823a93c65365d08bce8","is_oa":true,"landing_page_url":"https://doaj.org/article/7a0de29ce7294823a93c65365d08bce8","pdf_url":null,"source":{"id":"https://openalex.org/S112646816","display_name":"SHILAP Revista de lepidopterolog\u00eda","issn_l":"0300-5267","issn":["0300-5267","2340-4078"],"is_oa":true,"is_in_doaj":true,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Algorithms, Vol 14, Iss 4, p 126 (2021)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/1999-4893/14/4/126/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/a14040126","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Algorithms; Volume 14; Issue 4; Pages: 126","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/a14040126","is_oa":true,"landing_page_url":"https://doi.org/10.3390/a14040126","pdf_url":"https://www.mdpi.com/1999-4893/14/4/126/pdf?version=1618927534","source":{"id":"https://openalex.org/S190629608","display_name":"Algorithms","issn_l":"1999-4893","issn":["1999-4893"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Algorithms","raw_type":"journal-article"},"sustainable_development_goals":[{"display_name":"Quality Education","score":0.49000000953674316,"id":"https://metadata.un.org/sdg/4"}],"awards":[{"id":"https://openalex.org/G6040236433","display_name":null,"funder_award_id":"075-02-2020-1537/1","funder_id":"https://openalex.org/F4320327494","funder_display_name":"Ministry of Science and Higher Education of the Russian Federation"}],"funders":[{"id":"https://openalex.org/F4320327494","display_name":"Ministry of Science and Higher Education of the Russian Federation","ror":null}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3155567177.pdf","grobid_xml":"https://content.openalex.org/works/W3155567177.grobid-xml"},"referenced_works_count":33,"referenced_works":["https://openalex.org/W44219933","https://openalex.org/W1482941134","https://openalex.org/W1523480252","https://openalex.org/W1967466192","https://openalex.org/W1980322817","https://openalex.org/W1986611537","https://openalex.org/W1988786215","https://openalex.org/W2020979852","https://openalex.org/W2021093608","https://openalex.org/W2035314514","https://openalex.org/W2036772106","https://openalex.org/W2042822833","https://openalex.org/W2047897736","https://openalex.org/W2052987310","https://openalex.org/W2080562208","https://openalex.org/W2084593271","https://openalex.org/W2086965829","https://openalex.org/W2093900036","https://openalex.org/W2111471998","https://openalex.org/W2147340135","https://openalex.org/W2273472658","https://openalex.org/W2280268851","https://openalex.org/W2739122762","https://openalex.org/W2906873011","https://openalex.org/W3013547149","https://openalex.org/W3081952542","https://openalex.org/W4242610849","https://openalex.org/W4252110646","https://openalex.org/W6601797564","https://openalex.org/W6741565245","https://openalex.org/W6757175743","https://openalex.org/W6775398635","https://openalex.org/W6782158136"],"related_works":["https://openalex.org/W4376140277","https://openalex.org/W2501849745","https://openalex.org/W3169685561","https://openalex.org/W2091985989","https://openalex.org/W4367055576","https://openalex.org/W2146997899","https://openalex.org/W2077383771","https://openalex.org/W1837592468","https://openalex.org/W2910653847","https://openalex.org/W2914983903"],"abstract_inverted_index":{"Binary":[0],"cube-free":[1,81],"language":[2,6],"and":[3,64,92,94,101],"ternary":[4,121],"square-free":[5,122],"are":[7],"two":[8,24],"\u201ccanonical\u201d":[9],"representatives":[10],"of":[11,15,22,38,54,86,113,119],"a":[12],"wide":[13],"class":[14],"languages":[16,25],"defined":[17],"by":[18],"avoidance":[19],"properties.":[20],"Each":[21],"these":[23,45],"can":[26],"be":[27],"viewed":[28],"as":[29],"an":[30,65],"infinite":[31,58],"binary":[32],"tree":[33,118],"reflecting":[34],"the":[35,49,52,75,80,83,95,106,111,117,120],"prefix":[36],"order":[37],"its":[39],"elements.":[40],"We":[41,60,103],"study":[42],"how":[43],"\u201chomogenious\u201d":[44],"trees":[46],"are,":[47],"analysing":[48],"following":[50,76],"parameter:":[51],"density":[53,85,97,112],"branching":[55,87,114],"nodes":[56],"along":[57],"paths.":[59],"present":[61],"combinatorial":[62],"results":[63,78],"efficient":[66],"search":[67],"algorithm,":[68],"which":[69],"together":[70],"allowed":[71],"us":[72],"to":[73],"get":[74],"numerical":[77],"for":[79],"language:":[82],"minimal":[84],"points":[88,115],"is":[89,98],"between":[90,99],"3509/9120\u22480.38476":[91],"13/29\u22480.44828,":[93],"maximal":[96],"0.72":[100],"67/93\u22480.72043.":[102],"also":[104],"prove":[105],"lower":[107],"bound":[108],"223/868\u22480.25691":[109],"on":[110],"in":[116],"language.":[123]},"counts_by_year":[{"year":2021,"cited_by_count":1}],"updated_date":"2026-03-13T16:22:10.518609","created_date":"2025-10-10T00:00:00"}