{"id":"https://openalex.org/W2166853092","doi":"https://doi.org/10.1137/100814998","title":"The Complexity of Distributions","display_name":"The Complexity of Distributions","publication_year":2012,"publication_date":"2012-01-01","ids":{"openalex":"https://openalex.org/W2166853092","doi":"https://doi.org/10.1137/100814998","mag":"2166853092"},"language":"en","primary_location":{"id":"doi:10.1137/100814998","is_oa":false,"landing_page_url":"https://doi.org/10.1137/100814998","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/A5072846764","display_name":"Emanuele Viola","orcid":"https://orcid.org/0000-0001-6091-1824"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Emanuele Viola","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5072846764"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":5.1831,"has_fulltext":false,"cited_by_count":44,"citation_normalized_percentile":{"value":0.9600716,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"41","issue":"1","first_page":"191","last_page":"218"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":1.0,"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":1.0,"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/T12072","display_name":"Machine Learning and Algorithms","score":0.9991000294685364,"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/T10237","display_name":"Cryptography and Data Security","score":0.9980999827384949,"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/combinatorics","display_name":"Combinatorics","score":0.746952474117279},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7358973026275635},{"id":"https://openalex.org/keywords/omega","display_name":"Omega","score":0.6423693895339966},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.5586336255073547},{"id":"https://openalex.org/keywords/hamming-weight","display_name":"Hamming weight","score":0.5538443922996521},{"id":"https://openalex.org/keywords/boolean-function","display_name":"Boolean function","score":0.5444941520690918},{"id":"https://openalex.org/keywords/distribution","display_name":"Distribution (mathematics)","score":0.5369837880134583},{"id":"https://openalex.org/keywords/binary-logarithm","display_name":"Binary logarithm","score":0.5258491635322571},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.511879563331604},{"id":"https://openalex.org/keywords/hamming-distance","display_name":"Hamming distance","score":0.5088655948638916},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5002472400665283},{"id":"https://openalex.org/keywords/tree","display_name":"Tree (set theory)","score":0.4481573700904846},{"id":"https://openalex.org/keywords/random-variable","display_name":"Random variable","score":0.43661054968833923},{"id":"https://openalex.org/keywords/hamming-code","display_name":"Hamming code","score":0.31112366914749146},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.18791934847831726},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.1288468837738037}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.746952474117279},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7358973026275635},{"id":"https://openalex.org/C2779557605","wikidata":"https://www.wikidata.org/wiki/Q9890","display_name":"Omega","level":2,"score":0.6423693895339966},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5586336255073547},{"id":"https://openalex.org/C63361517","wikidata":"https://www.wikidata.org/wiki/Q5645805","display_name":"Hamming weight","level":5,"score":0.5538443922996521},{"id":"https://openalex.org/C187455244","wikidata":"https://www.wikidata.org/wiki/Q942353","display_name":"Boolean function","level":2,"score":0.5444941520690918},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.5369837880134583},{"id":"https://openalex.org/C63553672","wikidata":"https://www.wikidata.org/wiki/Q581168","display_name":"Binary logarithm","level":2,"score":0.5258491635322571},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.511879563331604},{"id":"https://openalex.org/C193319292","wikidata":"https://www.wikidata.org/wiki/Q272172","display_name":"Hamming distance","level":2,"score":0.5088655948638916},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5002472400665283},{"id":"https://openalex.org/C113174947","wikidata":"https://www.wikidata.org/wiki/Q2859736","display_name":"Tree (set theory)","level":2,"score":0.4481573700904846},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.43661054968833923},{"id":"https://openalex.org/C73150493","wikidata":"https://www.wikidata.org/wiki/Q853922","display_name":"Hamming code","level":4,"score":0.31112366914749146},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.18791934847831726},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.1288468837738037},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C157125643","wikidata":"https://www.wikidata.org/wiki/Q884707","display_name":"Block code","level":3,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C57273362","wikidata":"https://www.wikidata.org/wiki/Q576722","display_name":"Decoding methods","level":2,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/100814998","is_oa":false,"landing_page_url":"https://doi.org/10.1137/100814998","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":[{"score":0.4399999976158142,"display_name":"Peace, Justice and strong institutions","id":"https://metadata.un.org/sdg/16"}],"awards":[{"id":"https://openalex.org/G4343776779","display_name":null,"funder_award_id":"CCF-0845003","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7695462847","display_name":"CAREER: New Pseudorandom Generators: Unconditional Results and Efficient Constructions (TOC)","funder_award_id":"0845003","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G848032724","display_name":null,"funder_award_id":"Science","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":21,"referenced_works":["https://openalex.org/W1964089073","https://openalex.org/W1984459117","https://openalex.org/W1986970296","https://openalex.org/W1998918799","https://openalex.org/W1999474022","https://openalex.org/W2014482556","https://openalex.org/W2014485295","https://openalex.org/W2019038806","https://openalex.org/W2023915883","https://openalex.org/W2034946612","https://openalex.org/W2050253549","https://openalex.org/W2060260518","https://openalex.org/W2110761203","https://openalex.org/W2111824166","https://openalex.org/W2141431275","https://openalex.org/W2144919274","https://openalex.org/W2148858749","https://openalex.org/W2155834086","https://openalex.org/W2164284862","https://openalex.org/W2988890240","https://openalex.org/W4251436657"],"related_works":["https://openalex.org/W2008960744","https://openalex.org/W2105471374","https://openalex.org/W2963549154","https://openalex.org/W2158117886","https://openalex.org/W1987803429","https://openalex.org/W1582340598","https://openalex.org/W1519122282","https://openalex.org/W2152162311","https://openalex.org/W1530654122","https://openalex.org/W2182731056"],"abstract_inverted_index":{"Complexity":[0],"theory":[1],"typically":[2],"studies":[3],"the":[4,28,61,107,131,149,154,160,179,184,199,222,237,312],"complexity":[5,29],"of":[6,17,30,113,151,162,175,187,194,206,244,251,270],"computing":[7],"a":[8,18,47,139,173,189,204,315],"function":[9,65],"$h(x)":[10],":":[11,67],"\\{0,":[12,15,68,71],"1\\}^m":[13],"\\to":[14,70],"1\\}^n$":[16,72],"given":[19,38],"input":[20,84],"x.":[21],"A":[22],"few":[23],"works":[24],"have":[25,320],"suggested":[26],"studying":[27],"generating\u2014or":[31],"sampling\u2014the":[32],"distribution":[33,97,109],"$h(x)$":[34],"for":[35,53,123,126,159,168,183,241],"uniform":[36,108,266],"x,":[37],"random":[39,163,288],"bits.":[40,232,289],"We":[41,118],"further":[42],"advocate":[43],"this":[44],"study,":[45],"with":[46,259,276],"new":[48],"emphasis":[49],"on":[50,81],"lower":[51,121,144,181],"bounds":[52,122,145,167,235,238],"restricted":[54],"computational":[55],"models.":[56],"Our":[57],"main":[58],"results":[59,158],"are":[60,213],"following:":[62],"(1)":[63,240],"Any":[64],"$f":[66],"1\\}^\\ell":[69],"such":[73],"that":[74],"(i)":[75],"each":[76,135],"output":[77,96],"bit":[78,136],"$f_i$":[79,137],"depends":[80],"$o(\\log":[82,218],"n)$":[83,219,231],"bits,":[85,220],"and":[86,129,254,273,294],"(ii)":[87],"$\\ell":[88],"\\le":[89],"\\log_2":[90,226],"\\binom{n}{\\alpha":[91,227],"n}":[92,228],"+":[93,229,279],"n^{0.99}$":[94],"has":[95],"$f(U)$":[98],"at":[99],"statistical":[100],"distance":[101],"$\\ge":[102,225],"1":[103],"-":[104],"1/n^{0.49}$":[105],"from":[106],"over":[110],"n-bit":[111],"strings":[112],"hamming":[114],"weight":[115],"$\\alpha":[116,196],"n$.":[117],"also":[119],"prove":[120],"generating":[124],"$(X,b(X))$":[125],"Boolean":[127],"b,":[128],"in":[130,133,198,239],"case":[132,200],"which":[134],"is":[138,203],"small-depth":[140],"decision":[141],"tree.":[142],"These":[143],"seem":[146],"to":[147],"be":[148,263],"first":[150,180],"their":[152],"kind;":[153],"proofs":[155],"use":[156],"anticoncentration":[157],"sum":[161],"variables.":[164],"(2)":[165],"Lower":[166],"succinct":[169],"data":[170],"structures.":[171],"As":[172],"corollary":[174],"(1),":[176],"we":[177],"obtain":[178],"bound":[182],"membership":[185],"problem":[186],"representing":[188],"set":[190],"$S":[191],"\\subseteq":[192],"[n]$":[193],"size":[195,253,272],"n$,":[197],"where":[201],"$1/\\alpha$":[202],"power":[205],"2:":[207],"If":[208],"queries":[209],"\u201c$i":[210],"\\in":[211],"S$?\u201d":[212],"answered":[214],"by":[215,265,314],"nonadaptively":[216],"probing":[217],"then":[221],"representation":[223],"uses":[224],"\\Omega(\\log":[230],"(3)":[233],"Upper":[234],"complementing":[236],"various":[242],"settings":[243],"parameters.":[245],"(4)":[246],"Uniform":[247],"randomized":[248,267],"$\\mathrm{AC}^0$":[249,268],"circuits":[250,269],"$\\mathrm{poly}(n)$":[252,271],"depth":[255,274,313],"$d":[256],"=":[257],"O(1)$":[258],"error":[260,277],"$\\epsilon$":[261],"can":[262],"simulated":[264],"$d+1$":[275],"$\\epsilon":[278],"o(1)$":[280],"using":[281],"$\\le":[282],"(\\log":[283],"n)^{O(":[284],"\\log":[285,286],"n)}$":[287],"Previous":[290],"derandomizations":[291],"[M.":[292],"Ajtai":[293],"A.":[295],"Wigderson,":[296],"Adv.":[297],"Comput.":[298],"Res.,":[299],"5":[300],"(1989),":[301],"pp.":[302,309],"199\u2013223],":[303],"[N.":[304],"Nisan,":[305],"Combinatorica,":[306],"11":[307],"(1991),":[308],"63\u201370]":[310],"increase":[311],"constant":[316],"factor,":[317],"or":[318],"else":[319],"poor":[321],"seed":[322],"length.":[323]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":4},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":5},{"year":2016,"cited_by_count":6},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":6},{"year":2012,"cited_by_count":4}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}
