{"id":"https://openalex.org/W2068244353","doi":"https://doi.org/10.1137/110850104","title":"Optimal Multivalued Shattering","display_name":"Optimal Multivalued Shattering","publication_year":2012,"publication_date":"2012-01-01","ids":{"openalex":"https://openalex.org/W2068244353","doi":"https://doi.org/10.1137/110850104","mag":"2068244353"},"language":"en","primary_location":{"is_oa":false,"landing_page_url":"https://doi.org/10.1137/110850104","pdf_url":null,"source":{"id":"https://openalex.org/S117679221","display_name":"SIAM Journal on Discrete Mathematics","issn_l":"0895-4801","issn":["0895-4801","1095-7146"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false},"type":"article","type_crossref":"journal-article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"http://arxiv.org/pdf/1109.1748","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5023439517","display_name":"Zolt\u00e1n F\u00fcredi","orcid":"https://orcid.org/0000-0001-7335-991X"},"institutions":[{"id":"https://openalex.org/I7597260","display_name":"Hungarian Academy of Sciences","ror":"https://ror.org/02ks8qq67","country_code":"HU","type":"government","lineage":["https://openalex.org/I7597260"]}],"countries":["HU"],"is_corresponding":false,"raw_author_name":"Zolt\u00e1n F\u00fcredi","raw_affiliation_strings":["Hungarian Academy of Sciences"],"affiliations":[{"raw_affiliation_string":"Hungarian Academy of Sciences","institution_ids":["https://openalex.org/I7597260"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5012184705","display_name":"Attila Sali","orcid":"https://orcid.org/0000-0002-4837-6360"},"institutions":[{"id":"https://openalex.org/I7597260","display_name":"Hungarian Academy of Sciences","ror":"https://ror.org/02ks8qq67","country_code":"HU","type":"government","lineage":["https://openalex.org/I7597260"]}],"countries":["HU"],"is_corresponding":false,"raw_author_name":"Attila Sali","raw_affiliation_strings":["Hungarian Academy of Sciences"],"affiliations":[{"raw_affiliation_string":"Hungarian Academy of Sciences","institution_ids":["https://openalex.org/I7597260"]}]}],"institution_assertions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.576,"has_fulltext":true,"fulltext_origin":"ngrams","cited_by_count":6,"citation_normalized_percentile":{"value":0.753321,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":81,"max":82},"biblio":{"volume":"26","issue":"2","first_page":"737","last_page":"744"},"is_retracted":false,"is_paratext":false,"primary_topic":{"id":"https://openalex.org/T11567","display_name":"semigroups and automata theory","score":0.9995,"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":0.9995,"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/T11797","display_name":"graph theory and CDMA systems","score":0.9995,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9994,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/exponent","display_name":"Exponent","score":0.6046796}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8242784},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7824528},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.6054931},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.6046796},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.469492},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4339304},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.10287261},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.07456526},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"is_oa":false,"landing_page_url":"https://doi.org/10.1137/110850104","pdf_url":null,"source":{"id":"https://openalex.org/S117679221","display_name":"SIAM Journal on Discrete Mathematics","issn_l":"0895-4801","issn":["0895-4801","1095-7146"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false},{"is_oa":true,"landing_page_url":"http://arxiv.org/abs/1109.1748","pdf_url":"http://arxiv.org/pdf/1109.1748","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":["Cornell University"],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false},{"is_oa":true,"landing_page_url":null,"pdf_url":"http://arxiv.org/pdf/1109.1748.pdf","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":["Cornell University"],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false}],"best_oa_location":{"is_oa":true,"landing_page_url":"http://arxiv.org/abs/1109.1748","pdf_url":"http://arxiv.org/pdf/1109.1748","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":["Cornell University"],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false},"sustainable_development_goals":[],"grants":[],"datasets":[],"versions":[],"referenced_works_count":13,"referenced_works":["https://openalex.org/W1971361630","https://openalex.org/W1971662168","https://openalex.org/W1973663311","https://openalex.org/W1978227150","https://openalex.org/W1981207991","https://openalex.org/W1990696900","https://openalex.org/W2020913456","https://openalex.org/W2044230505","https://openalex.org/W2051905988","https://openalex.org/W2114935567","https://openalex.org/W4213201446","https://openalex.org/W4235786133","https://openalex.org/W4238284510"],"related_works":["https://openalex.org/W4287865632","https://openalex.org/W4286787881","https://openalex.org/W3194077685","https://openalex.org/W3049435653","https://openalex.org/W2950599922","https://openalex.org/W2467119940","https://openalex.org/W2391509332","https://openalex.org/W2382762871","https://openalex.org/W2152758797","https://openalex.org/W2062219402"],"abstract_inverted_index":{"We":[0,59,116],"have":[1],"found":[2],"a":[3,25,35,42,103,118],"general":[4],"extension":[5],"of":[6,38,44,51,88],"the":[7,49,69,125],"celebrated":[8],"Sauer,":[9],"Perles":[10],"and":[11,14,77,98],"Shelah,":[12],"Vapnik":[13],"Chervonenkis":[15],"result":[16],"from":[17],"0-1":[18],"sequences":[19],"to":[20,53],"k-ary":[21,36],"codes":[22],"still":[23],"giving":[24],"polynomial":[26],"bound.":[27],"Let":[28],"$\\mathcal{C}\\subseteq":[29],"\\{":[30],"0,1,\\dots,":[31],"k-1":[32],"\\}^n$":[33],"be":[34],"code":[37],"length":[39],"n.":[40],"For":[41],"subset":[43],"coordinates":[45,75],"$S\\subset":[46],"\\{1,2,\\ldots":[47],",n\\}$,":[48],"projection":[50],"$\\mathcal{C}$":[52,62,81],"S":[54,64],"is":[55,124],"denoted":[56],"by":[57],"$\\mathcal{C}\\vert_S$.":[58],"say":[60],"that":[61,80,108,121],"$(i,j)$-shatters":[63],"if":[65],"$\\mathcal{C}\\vert_S$":[66],"contains":[67],"all":[68],"$2^{|S|}$":[70],"distinct":[71],"vectors":[72],"(codewords)":[73],"with":[74],"i":[76],"j.":[78],"Suppose":[79],"does":[82],"not":[83],"$(i,j)$-shatter":[84],"any":[85],"coordinate":[86],"set":[87],"size":[89],"$s_{i,j}\\geq":[90],"1$":[91],"for":[92],"every":[93],"$0\\leq":[94],"i<":[95],"j\\leq":[96],"k-1$,":[97],"let":[99],"$p=\\sum":[100],"(s_{i,j}-1)$.":[101],"Using":[102],"natural":[104],"induction":[105],"we":[106],"prove":[107],"$|{\\mathcal":[109],"C}|\\leq":[110],"O(n^p)":[111],"$":[112],"as":[113],"$n\\to":[114],"\\infty$.":[115],"give":[117],"construction":[119],"showing":[120],"this":[122],"exponent":[123],"best":[126],"possible.":[127],"Several":[128],"open":[129],"problems":[130],"are":[131],"mentioned.":[132]},"cited_by_api_url":"https://api.openalex.org/works?filter=cites:W2068244353","counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":1}],"updated_date":"2024-12-17T14:18:21.491188","created_date":"2016-06-24"}