{"id":"https://openalex.org/W2153680565","doi":"https://doi.org/10.1137/090764190","title":"Explicit Construction of a Small $\\epsilon$-Net for Linear Threshold Functions","display_name":"Explicit Construction of a Small $\\epsilon$-Net for Linear Threshold Functions","publication_year":2010,"publication_date":"2010-01-01","ids":{"openalex":"https://openalex.org/W2153680565","doi":"https://doi.org/10.1137/090764190","mag":"2153680565"},"language":"en","primary_location":{"id":"doi:10.1137/090764190","is_oa":false,"landing_page_url":"https://doi.org/10.1137/090764190","pdf_url":null,"source":{"id":"https://openalex.org/S153560523","display_name":"SIAM Journal on Computing","issn_l":"0097-5397","issn":["0097-5397","1095-7111"],"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":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Computing","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5010842984","display_name":"Yuval Rabani","orcid":"https://orcid.org/0000-0001-7772-2544"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Yuval Rabani","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5004182247","display_name":"Amir Shpilka","orcid":"https://orcid.org/0000-0003-2384-425X"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Amir Shpilka","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5010842984"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":3.7767,"has_fulltext":false,"cited_by_count":28,"citation_normalized_percentile":{"value":0.93659308,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"39","issue":"8","first_page":"3501","last_page":"3520"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","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"}},"topics":[{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","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/T12162","display_name":"Cellular Automata and Applications","score":0.9966999888420105,"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/T10796","display_name":"Cooperative Communication and Network Coding","score":0.9965999722480774,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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/mathematics","display_name":"Mathematics","score":0.7699742913246155},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7329593300819397},{"id":"https://openalex.org/keywords/exponent","display_name":"Exponent","score":0.6488898992538452},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.6438726782798767},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.6113085746765137},{"id":"https://openalex.org/keywords/unit-cube","display_name":"Unit cube","score":0.5694684982299805},{"id":"https://openalex.org/keywords/bch-code","display_name":"BCH code","score":0.5419356226921082},{"id":"https://openalex.org/keywords/radius","display_name":"RADIUS","score":0.5320951342582703},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.49787259101867676},{"id":"https://openalex.org/keywords/cube","display_name":"Cube (algebra)","score":0.48642921447753906},{"id":"https://openalex.org/keywords/binary-number","display_name":"Binary number","score":0.4828394055366516},{"id":"https://openalex.org/keywords/time-complexity","display_name":"Time complexity","score":0.4410296678543091},{"id":"https://openalex.org/keywords/unit","display_name":"Unit (ring theory)","score":0.4273056983947754},{"id":"https://openalex.org/keywords/decoding-methods","display_name":"Decoding methods","score":0.1283743977546692},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.11197951436042786},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.08474323153495789},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.07061982154846191}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7699742913246155},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7329593300819397},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.6488898992538452},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.6438726782798767},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.6113085746765137},{"id":"https://openalex.org/C60432849","wikidata":"https://www.wikidata.org/wiki/Q2072112","display_name":"Unit cube","level":2,"score":0.5694684982299805},{"id":"https://openalex.org/C42276685","wikidata":"https://www.wikidata.org/wiki/Q795705","display_name":"BCH code","level":3,"score":0.5419356226921082},{"id":"https://openalex.org/C178635117","wikidata":"https://www.wikidata.org/wiki/Q747499","display_name":"RADIUS","level":2,"score":0.5320951342582703},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.49787259101867676},{"id":"https://openalex.org/C53051483","wikidata":"https://www.wikidata.org/wiki/Q861555","display_name":"Cube (algebra)","level":2,"score":0.48642921447753906},{"id":"https://openalex.org/C48372109","wikidata":"https://www.wikidata.org/wiki/Q3913","display_name":"Binary number","level":2,"score":0.4828394055366516},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.4410296678543091},{"id":"https://openalex.org/C122637931","wikidata":"https://www.wikidata.org/wiki/Q118084","display_name":"Unit (ring theory)","level":2,"score":0.4273056983947754},{"id":"https://openalex.org/C57273362","wikidata":"https://www.wikidata.org/wiki/Q576722","display_name":"Decoding methods","level":2,"score":0.1283743977546692},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.11197951436042786},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.08474323153495789},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.07061982154846191},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","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},{"id":"https://openalex.org/C145420912","wikidata":"https://www.wikidata.org/wiki/Q853077","display_name":"Mathematics education","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/C38652104","wikidata":"https://www.wikidata.org/wiki/Q3510521","display_name":"Computer security","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/090764190","is_oa":false,"landing_page_url":"https://doi.org/10.1137/090764190","pdf_url":null,"source":{"id":"https://openalex.org/S153560523","display_name":"SIAM Journal on Computing","issn_l":"0097-5397","issn":["0097-5397","1095-7111"],"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":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Computing","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":27,"referenced_works":["https://openalex.org/W21386869","https://openalex.org/W1972939134","https://openalex.org/W1988106539","https://openalex.org/W2002645420","https://openalex.org/W2019807548","https://openalex.org/W2052207834","https://openalex.org/W2055391790","https://openalex.org/W2055656009","https://openalex.org/W2068769655","https://openalex.org/W2068871408","https://openalex.org/W2077326980","https://openalex.org/W2092509105","https://openalex.org/W2098313059","https://openalex.org/W2114598336","https://openalex.org/W2126274204","https://openalex.org/W2129638195","https://openalex.org/W2150482801","https://openalex.org/W2151089248","https://openalex.org/W2165621523","https://openalex.org/W2167816765","https://openalex.org/W2296616510","https://openalex.org/W2611007065","https://openalex.org/W2623310026","https://openalex.org/W4235946010","https://openalex.org/W4250955649","https://openalex.org/W4285719527","https://openalex.org/W4385965835"],"related_works":["https://openalex.org/W2031281829","https://openalex.org/W4254427182","https://openalex.org/W2113702600","https://openalex.org/W1999568846","https://openalex.org/W4255848923","https://openalex.org/W2040634226","https://openalex.org/W2480729894","https://openalex.org/W2122070455","https://openalex.org/W2952361412","https://openalex.org/W2285875724"],"abstract_inverted_index":{"We":[0],"give":[1],"explicit":[2],"constructions":[3,42],"of":[4,21,37,53,71,85,98,108],"$\\epsilon$-nets":[5],"for":[6,88],"linear":[7],"threshold":[8],"functions":[9],"on":[10,15],"the":[11,16,22,28,35,51,54,56,72,95],"binary":[12,73],"cube":[13,74],"and":[14,31,81],"unit":[17],"sphere.":[18],"The":[19],"size":[20,77],"constructed":[23],"nets":[24],"is":[25],"polynomial":[26,78],"in":[27,32,79],"dimension":[29],"n":[30,80],"$\\frac{1}{\\epsilon}$.":[33],"To":[34],"best":[36],"our":[38],"knowledge":[39],"no":[40],"such":[41],"were":[43],"previously":[44],"known.":[45],"Our":[46],"results":[47],"match,":[48],"up":[49],"to":[50],"exponent":[52],"polynomial,":[55],"bounds":[57],"that":[58,75,102],"are":[59],"achieved":[60],"by":[61],"probabilistic":[62],"arguments.":[63],"As":[64],"a":[65,82,105],"corollary":[66],"we":[67],"also":[68],"construct":[69],"subsets":[70],"have":[76],"covering":[83,106],"radius":[84,107],"$\\frac{n}{2}-c\\sqrt{n\\log":[86],"n}$":[87],"any":[89],"constant":[90],"c.":[91],"This":[92],"improves":[93],"upon":[94],"well-known":[96],"construction":[97],"dual":[99],"BCH":[100],"codes":[101],"guarantee":[103],"only":[104],"$\\frac{n}{2}-c\\sqrt{n}$.":[109]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":3},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":4},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":4},{"year":2012,"cited_by_count":2}],"updated_date":"2026-04-04T08:04:53.788161","created_date":"2025-10-10T00:00:00"}
