{"id":"https://openalex.org/W2782168774","doi":"https://doi.org/10.1137/16m1061096","title":"Approximate Polytope Membership Queries","display_name":"Approximate Polytope Membership Queries","publication_year":2018,"publication_date":"2018-01-01","ids":{"openalex":"https://openalex.org/W2782168774","doi":"https://doi.org/10.1137/16m1061096","mag":"2782168774"},"language":"en","primary_location":{"id":"doi:10.1137/16m1061096","is_oa":false,"landing_page_url":"https://doi.org/10.1137/16m1061096","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":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1604.01183","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"Sunil Arya","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Sunil Arya","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":null,"display_name":"Guilherme D. da Fonseca","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Guilherme D. da Fonseca","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":null,"display_name":"David M. Mount","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"David M. Mount","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.2145,"has_fulltext":false,"cited_by_count":2,"citation_normalized_percentile":{"value":0.52027762,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"47","issue":"1","first_page":"1","last_page":"51"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9997000098228455,"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9997000098228455,"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/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"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/T12072","display_name":"Machine Learning and Algorithms","score":0.9977999925613403,"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/polytope","display_name":"Polytope","score":0.7742000222206116},{"id":"https://openalex.org/keywords/intersection","display_name":"Intersection (aeronautics)","score":0.5949000120162964},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.5684000253677368},{"id":"https://openalex.org/keywords/euclidean-space","display_name":"Euclidean space","score":0.5357999801635742},{"id":"https://openalex.org/keywords/convex-body","display_name":"Convex body","score":0.49810001254081726},{"id":"https://openalex.org/keywords/convex-polytope","display_name":"Convex polytope","score":0.4864000082015991},{"id":"https://openalex.org/keywords/simple","display_name":"Simple (philosophy)","score":0.46560001373291016},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.4153999984264374},{"id":"https://openalex.org/keywords/approximation-algorithm","display_name":"Approximation algorithm","score":0.4138999879360199}],"concepts":[{"id":"https://openalex.org/C145691206","wikidata":"https://www.wikidata.org/wiki/Q747980","display_name":"Polytope","level":2,"score":0.7742000222206116},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6714000105857849},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6672999858856201},{"id":"https://openalex.org/C64543145","wikidata":"https://www.wikidata.org/wiki/Q162942","display_name":"Intersection (aeronautics)","level":2,"score":0.5949000120162964},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.5684000253677368},{"id":"https://openalex.org/C186450821","wikidata":"https://www.wikidata.org/wiki/Q17295","display_name":"Euclidean space","level":2,"score":0.5357999801635742},{"id":"https://openalex.org/C134912446","wikidata":"https://www.wikidata.org/wiki/Q5166516","display_name":"Convex body","level":4,"score":0.49810001254081726},{"id":"https://openalex.org/C110943637","wikidata":"https://www.wikidata.org/wiki/Q4129097","display_name":"Convex polytope","level":5,"score":0.4864000082015991},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.46560001373291016},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.4153999984264374},{"id":"https://openalex.org/C148764684","wikidata":"https://www.wikidata.org/wiki/Q621751","display_name":"Approximation algorithm","level":2,"score":0.4138999879360199},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.4066999852657318},{"id":"https://openalex.org/C29123130","wikidata":"https://www.wikidata.org/wiki/Q874709","display_name":"Computational geometry","level":2,"score":0.39820000529289246},{"id":"https://openalex.org/C206304794","wikidata":"https://www.wikidata.org/wiki/Q4916482","display_name":"Birkhoff polytope","level":5,"score":0.39100000262260437},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3695000112056732},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.3682999908924103},{"id":"https://openalex.org/C206194317","wikidata":"https://www.wikidata.org/wiki/Q1138624","display_name":"Convex hull","level":3,"score":0.367000013589859},{"id":"https://openalex.org/C162319229","wikidata":"https://www.wikidata.org/wiki/Q175263","display_name":"Data structure","level":2,"score":0.3569999933242798},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.3447999954223633},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.34220001101493835},{"id":"https://openalex.org/C182964748","wikidata":"https://www.wikidata.org/wiki/Q208216","display_name":"Triangle inequality","level":2,"score":0.33219999074935913},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.304500013589859},{"id":"https://openalex.org/C110202963","wikidata":"https://www.wikidata.org/wiki/Q1783542","display_name":"Convex geometry","level":5,"score":0.30390000343322754},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.3034999966621399},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.29330000281333923},{"id":"https://openalex.org/C120174047","wikidata":"https://www.wikidata.org/wiki/Q847073","display_name":"Euclidean distance","level":2,"score":0.288100004196167},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.28029999136924744},{"id":"https://openalex.org/C49870271","wikidata":"https://www.wikidata.org/wiki/Q193657","display_name":"Convex set","level":4,"score":0.2759999930858612},{"id":"https://openalex.org/C108005400","wikidata":"https://www.wikidata.org/wiki/Q1305598","display_name":"Facility location problem","level":2,"score":0.2615000009536743},{"id":"https://openalex.org/C111110010","wikidata":"https://www.wikidata.org/wiki/Q2627315","display_name":"Convex combination","level":4,"score":0.2531000077724457}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1137/16m1061096","is_oa":false,"landing_page_url":"https://doi.org/10.1137/16m1061096","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"},{"id":"pmh:oai:arXiv.org:1604.01183","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1604.01183","pdf_url":"https://arxiv.org/pdf/1604.01183","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":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:HAL:hal-01890054v1","is_oa":true,"landing_page_url":"https://hal.science/hal-01890054","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"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":"SIAM Journal on Computing, 2018, 47 (1), pp.1 - 51. &#x27E8;10.1137/16M1061096&#x27E9;","raw_type":"Journal articles"},{"id":"pmh:oai:repository.hkust.edu.hk:1783.1-89405","is_oa":false,"landing_page_url":"http://repository.hkust.edu.hk/ir/Record/1783.1-89405","pdf_url":null,"source":{"id":"https://openalex.org/S4306401796","display_name":"Rare & Special e-Zone (The Hong Kong University of Science and Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I200769079","host_organization_name":"Hong Kong University of Science and Technology","host_organization_lineage":["https://openalex.org/I200769079"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1604.01183","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1604.01183","pdf_url":"https://arxiv.org/pdf/1604.01183","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":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G4121092108","display_name":null,"funder_award_id":"CCF-1618866","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G430242359","display_name":"Approximate geometric queries: Beyond Euclidean space","funder_award_id":"610108","funder_id":"https://openalex.org/F4320321592","funder_display_name":"Research Grants Council, University Grants Committee"},{"id":"https://openalex.org/G6860628828","display_name":"Faster approximation algorithms for proximity problems","funder_award_id":"16200014","funder_id":"https://openalex.org/F4320321592","funder_display_name":"Research Grants Council, University Grants Committee"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320321592","display_name":"Research Grants Council, University Grants Committee","ror":"https://ror.org/00djwmt25"},{"id":"https://openalex.org/F4320322025","display_name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico","ror":"https://ror.org/03swz6y49"},{"id":"https://openalex.org/F4320322749","display_name":"Funda\u00e7\u00e3o Carlos Chagas Filho de Amparo \u00e0 Pesquisa do Estado do Rio de Janeiro","ror":"https://ror.org/03kk0s825"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":24,"referenced_works":["https://openalex.org/W1981313592","https://openalex.org/W1996036165","https://openalex.org/W2006109604","https://openalex.org/W2007273639","https://openalex.org/W2016400774","https://openalex.org/W2037318282","https://openalex.org/W2039289489","https://openalex.org/W2042800588","https://openalex.org/W2047308801","https://openalex.org/W2050356354","https://openalex.org/W2052948190","https://openalex.org/W2058127000","https://openalex.org/W2068769655","https://openalex.org/W2075603209","https://openalex.org/W2079872371","https://openalex.org/W2085121963","https://openalex.org/W2086591901","https://openalex.org/W2088397521","https://openalex.org/W2103893986","https://openalex.org/W2139212933","https://openalex.org/W2151942047","https://openalex.org/W2218835040","https://openalex.org/W2220451813","https://openalex.org/W4248484045"],"related_works":[],"abstract_inverted_index":{"In":[0,184],"the":[1,14,56,64,81,91,136,144,164,263,271,310,313],"polytope":[2,7,100,109,258],"membership":[3,101,259],"problem,":[4],"a":[5,21,170,225,233,293],"convex":[6,226],"$K$":[8,19,87],"in":[9,47,135,281,312],"$\\mathbb{R}^d$":[10],"is":[11,16,34,71,84,88,133,146,215,224,247,289],"given,":[12],"and":[13,86,120,123,143,200],"objective":[15],"to":[17,36,68,98,141,150,162,168,249,256,270,291],"preprocess":[18],"into":[20],"data":[22,193],"structure":[23,194],"so":[24],"that,":[25],"given":[26],"any":[27,187],"query":[28,57,151,196,294],"point":[29],"$q":[30,40],"\\in":[31,41],"\\mathbb{R}^d$,":[32],"it":[33,246,288],"possible":[35,248,290],"determine":[37],"efficiently":[38],"whether":[39,211],"K$.":[42],"We":[43,78,153,207],"consider":[44],"this":[45,182,192,212,266],"problem":[46],"an":[48,52,177],"approximate":[49,99,251,257,277],"setting.":[50],"Given":[51],"approximation":[53,110],"parameter":[54],"$\\varepsilon$,":[55],"can":[58],"be":[59],"answered":[60],"either":[61],"way":[62],"if":[63],"distance":[65],"from":[66],"$q$":[67],"$K$'s":[69,76],"boundary":[70],"at":[72,236],"most":[73],"$\\varepsilon$":[74],"times":[75],"diameter.":[77],"assume":[79],"that":[80,179,222,229,245,287],"dimension":[82],"$d$":[83],"fixed,":[85],"presented":[89],"as":[90],"intersection":[92],"of":[93,107,235,296],"$n$":[94],"halfspaces.":[95],"Previous":[96],"solutions":[97],"were":[102],"based":[103],"on":[104],"straightforward":[105],"applications":[106],"classic":[108],"techniques":[111,167],"by":[112,308],"Dudley":[113],"[Approx.":[114],"Theory,":[115],"10":[116],"(1974),":[117],"pp.":[118,129],"227--236]":[119],"Bentley,":[121],"Faust,":[122],"Preparata":[124],"[Commun.":[125],"ACM,":[126],"25":[127],"(1982),":[128],"64--68].":[130],"The":[131],"former":[132],"optimal":[134,147],"worst":[137],"case":[138],"with":[139,148,262,302],"respect":[140,149],"space,":[142],"latter":[145],"time.":[152],"present":[154,176],"four":[155],"main":[156],"results.":[157],"First,":[158],"we":[159,175,285],"show":[160,286],"how":[161],"combine":[163],"two":[165],"above":[166,264],"obtain":[169],"simple":[171],"space-time":[172,274],"trade-off.":[173,183],"Second,":[174],"algorithm":[178,231],"dramatically":[180],"improves":[181],"particular,":[185],"for":[186,276],"constant":[188],"$\\alpha":[189],"\\ge":[190],"4$,":[191],"achieves":[195,232],"time":[197,295],"roughly":[198,202,297,304],"$O(1/\\varepsilon^{(d-1)/\\alpha})$":[199],"space":[201,213,234,303,314],"$O(1/\\varepsilon^{(d-1)(1":[203],"-":[204],"\\Omega(\\log":[205],"\\alpha)/\\alpha)})$.":[206],"do":[208],"not":[209],"know":[210],"bound":[214],"tight,":[216],"but":[217],"our":[218,230],"third":[219],"result":[220,243],"shows":[221,244],"there":[223],"body":[227],"such":[228],"least":[237],"$\\Omega(":[238],"1/\\varepsilon^{(d-1)(1-O(\\sqrt{\\alpha})/\\alpha}":[239],")$.":[240],"Our":[241],"fourth":[242],"reduce":[250],"Euclidean":[252],"nearest":[253,278],"neighbor":[254,279],"searching":[255,280],"queries.":[260],"Combined":[261],"results,":[265],"provides":[267],"significant":[268],"improvements":[269],"best":[272],"known":[273],"trade-offs":[275],"$\\mathbb{R}^d$.":[282],"For":[283],"example,":[284],"achieve":[292],"$O(\\log":[298],"n":[299],"+":[300],"1/\\varepsilon^{d/4})$":[301],"$O(n/\\varepsilon^{d/4})$,":[305],"thus":[306],"reducing":[307],"half":[309],"exponent":[311],"bound.":[315]},"counts_by_year":[{"year":2022,"cited_by_count":1},{"year":2019,"cited_by_count":1}],"updated_date":"2026-06-22T08:00:12.763002","created_date":"2018-01-12T00:00:00"}
