{"id":"https://openalex.org/W4226019255","doi":"https://doi.org/10.1007/s00453-022-01091-y","title":"Algorithms and Lower Bounds for Comparator Circuits from Shrinkage","display_name":"Algorithms and Lower Bounds for Comparator Circuits from Shrinkage","publication_year":2023,"publication_date":"2023-01-13","ids":{"openalex":"https://openalex.org/W4226019255","doi":"https://doi.org/10.1007/s00453-022-01091-y"},"language":"en","primary_location":{"id":"doi:10.1007/s00453-022-01091-y","is_oa":true,"landing_page_url":"http://dx.doi.org/10.1007/s00453-022-01091-y","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00453-022-01091-y.pdf","source":{"id":"https://openalex.org/S89324355","display_name":"Algorithmica","issn_l":"0178-4617","issn":["0178-4617","1432-0541"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Algorithmica","raw_type":"journal-article"},"type":"preprint","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s00453-022-01091-y.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5009822741","display_name":"Bruno Pasqualotto Cavalar","orcid":"https://orcid.org/0000-0002-0458-8767"},"institutions":[{"id":"https://openalex.org/I39555362","display_name":"University of Warwick","ror":"https://ror.org/01a77tt86","country_code":"GB","type":"education","lineage":["https://openalex.org/I39555362"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Bruno P. Cavalar","raw_affiliation_strings":["Department of Computer Science, University of Warwick, Coventry, UK"],"affiliations":[{"raw_affiliation_string":"Department of Computer Science, University of Warwick, Coventry, UK","institution_ids":["https://openalex.org/I39555362"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5032937708","display_name":"Zhenjian Lu","orcid":"https://orcid.org/0009-0007-3990-4751"},"institutions":[{"id":"https://openalex.org/I40120149","display_name":"University of Oxford","ror":"https://ror.org/052gg0110","country_code":"GB","type":"education","lineage":["https://openalex.org/I40120149"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Zhenjian Lu","raw_affiliation_strings":["Department of Computer Science, University of Oxford, Oxford, UK"],"affiliations":[{"raw_affiliation_string":"Department of Computer Science, University of Oxford, Oxford, UK","institution_ids":["https://openalex.org/I40120149"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5009822741"],"corresponding_institution_ids":["https://openalex.org/I39555362"],"apc_list":{"value":2290,"currency":"EUR","value_usd":2890},"apc_paid":{"value":2290,"currency":"EUR","value_usd":2890},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.00846276,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"85","issue":"7","first_page":"2131","last_page":"2155"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9991000294685364,"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":0.9991000294685364,"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":0.9861000180244446,"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/T11567","display_name":"semigroups and automata theory","score":0.9821000099182129,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/comparator","display_name":"Comparator","score":0.8496827483177185},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6754510998725891},{"id":"https://openalex.org/keywords/nondeterministic-algorithm","display_name":"Nondeterministic algorithm","score":0.6596229076385498},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.6351494789123535},{"id":"https://openalex.org/keywords/binary-logarithm","display_name":"Binary logarithm","score":0.608107328414917},{"id":"https://openalex.org/keywords/omega","display_name":"Omega","score":0.6021607518196106},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5763067603111267},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.556492805480957},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4695618450641632},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4623362720012665},{"id":"https://openalex.org/keywords/electronic-circuit","display_name":"Electronic circuit","score":0.418140172958374},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3728918433189392},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.2041052281856537},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.11978623270988464}],"concepts":[{"id":"https://openalex.org/C155745195","wikidata":"https://www.wikidata.org/wiki/Q1164179","display_name":"Comparator","level":3,"score":0.8496827483177185},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6754510998725891},{"id":"https://openalex.org/C176181172","wikidata":"https://www.wikidata.org/wiki/Q3490301","display_name":"Nondeterministic algorithm","level":2,"score":0.6596229076385498},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.6351494789123535},{"id":"https://openalex.org/C63553672","wikidata":"https://www.wikidata.org/wiki/Q581168","display_name":"Binary logarithm","level":2,"score":0.608107328414917},{"id":"https://openalex.org/C2779557605","wikidata":"https://www.wikidata.org/wiki/Q9890","display_name":"Omega","level":2,"score":0.6021607518196106},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5763067603111267},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.556492805480957},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4695618450641632},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4623362720012665},{"id":"https://openalex.org/C134146338","wikidata":"https://www.wikidata.org/wiki/Q1815901","display_name":"Electronic circuit","level":2,"score":0.418140172958374},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3728918433189392},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.2041052281856537},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.11978623270988464},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C165801399","wikidata":"https://www.wikidata.org/wiki/Q25428","display_name":"Voltage","level":2,"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/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1007/s00453-022-01091-y","is_oa":true,"landing_page_url":"http://dx.doi.org/10.1007/s00453-022-01091-y","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00453-022-01091-y.pdf","source":{"id":"https://openalex.org/S89324355","display_name":"Algorithmica","issn_l":"0178-4617","issn":["0178-4617","1432-0541"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Algorithmica","raw_type":"journal-article"},{"id":"pmh:oai:wrap.warwick.ac.uk:174025","is_oa":true,"landing_page_url":null,"pdf_url":"https://wrap.warwick.ac.uk/174025/1/s00453-022-01091-y.pdf","source":{"id":"https://openalex.org/S4306400665","display_name":"Warwick Research Archive Portal (University of Warwick)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I39555362","host_organization_name":"University of Warwick","host_organization_lineage":["https://openalex.org/I39555362"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Journal Article"}],"best_oa_location":{"id":"doi:10.1007/s00453-022-01091-y","is_oa":true,"landing_page_url":"http://dx.doi.org/10.1007/s00453-022-01091-y","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00453-022-01091-y.pdf","source":{"id":"https://openalex.org/S89324355","display_name":"Algorithmica","issn_l":"0178-4617","issn":["0178-4617","1432-0541"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Algorithmica","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1420847930","display_name":null,"funder_award_id":"URF\\R1\\19105","funder_id":"https://openalex.org/F4320320006","funder_display_name":"Royal Society"},{"id":"https://openalex.org/G2674840903","display_name":null,"funder_award_id":"Fellowship","funder_id":"https://openalex.org/F4320320006","funder_display_name":"Royal Society"},{"id":"https://openalex.org/G5016171127","display_name":null,"funder_award_id":"URF\\R1\\191059","funder_id":"https://openalex.org/F4320320006","funder_display_name":"Royal Society"},{"id":"https://openalex.org/G6959233825","display_name":null,"funder_award_id":"000000","funder_id":"https://openalex.org/F4320320279","funder_display_name":"University of Warwick"}],"funders":[{"id":"https://openalex.org/F4320320006","display_name":"Royal Society","ror":"https://ror.org/03wnrjx87"},{"id":"https://openalex.org/F4320320279","display_name":"University of Warwick","ror":"https://ror.org/01a77tt86"}],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4226019255.pdf"},"referenced_works_count":24,"referenced_works":["https://openalex.org/W1995240561","https://openalex.org/W2004130530","https://openalex.org/W2016263640","https://openalex.org/W2030586582","https://openalex.org/W2053658780","https://openalex.org/W2059387708","https://openalex.org/W2063844956","https://openalex.org/W2076458424","https://openalex.org/W2083820240","https://openalex.org/W2165293955","https://openalex.org/W2173502887","https://openalex.org/W2490907619","https://openalex.org/W2565821653","https://openalex.org/W2585952665","https://openalex.org/W2888250735","https://openalex.org/W2938685469","https://openalex.org/W2965961151","https://openalex.org/W2971710618","https://openalex.org/W2997373829","https://openalex.org/W3023000450","https://openalex.org/W3034628227","https://openalex.org/W3044440032","https://openalex.org/W4205582340","https://openalex.org/W6629067008"],"related_works":["https://openalex.org/W1022019","https://openalex.org/W9459877","https://openalex.org/W58132005","https://openalex.org/W38306794","https://openalex.org/W18873129","https://openalex.org/W38382899","https://openalex.org/W44075917","https://openalex.org/W31007250","https://openalex.org/W58865375","https://openalex.org/W7548172"],"abstract_inverted_index":{"Abstract":[0],"In":[1],"this":[2,39],"paper,":[3],"we":[4,22,43,137],"initiate":[5],"the":[6,24,229,271],"study":[7],"of":[8,26,35,233,273,276],"average-case":[9,225],"complexity":[10],"and":[11,235],"circuit":[12,101,280],"analysis":[13],"algorithms":[14],"for":[15,38,98],"comparator":[16,100,279],"circuits.":[17],"Departing":[18],"from":[19],"previous":[20],"approaches,":[21],"exploit":[23],"technique":[25],"shrinkage":[27],"under":[28],"random":[29],"restrictions":[30],"to":[31],"obtain":[32],"a":[33,81,277],"variety":[34],"new":[36],"results":[37],"model.":[40],"Among":[41],"them,":[42],"show":[44],"Average-case":[45],"Lower":[46],"Bounds":[47],"For":[48],"every":[49,99],"$$k":[50,65],"=":[51],"k(n)$$":[52],"<mml:math":[53,69,86,111,153,243,259,289,319],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">":[54,70,87,112,154,244,260,290,320],"<mml:mrow>":[55,71,113,116,128,155,159,161,168,183,193,209,245,291,294,306,321,324,331,338],"<mml:mi>k</mml:mi>":[56,58,72,90,125,191,212,246,303,327],"<mml:mo>=</mml:mo>":[57,188,247],"<mml:mo>(</mml:mo>":[59,184,194,211],"<mml:mi>n</mml:mi>":[60,75,115,130,177,253,293,308,325],"<mml:mo>)</mml:mo>":[61,186,196,213],"</mml:mrow>":[62,76,118,131,134,167,179,187,197,214,218,222,255,296,309,312,328,337],"</mml:math>":[63,77,92,135,223,256,262,313],"with":[64,103,281],"\\geqslant":[66],"\\log":[67,240],"n$$":[68],"<mml:mo>\u2a7e</mml:mo>":[73],"<mml:mo>log</mml:mo>":[74,129,252,307],",":[78],"there":[79],"exists":[80],"polynomial-time":[82],"computable":[83],"function":[84],"$$f_k$$":[85],"<mml:msub>":[88,189],"<mml:mi>f</mml:mi>":[89,190],"</mml:msub>":[91,192],"on":[93],"n":[94],"bits":[95],"such":[96],"that,":[97],"C":[102],"at":[104,282],"most":[105,283],"$$n^{1.5}/O\\!\\left(":[106,284],"k\\cdot":[107,285],"\\sqrt{\\log":[108,286],"n}\\right)":[109,287],"$$":[110,242,288],"<mml:msup>":[114,171,207,292,322],"<mml:mn>1.5</mml:mn>":[117,295],"</mml:msup>":[119,178,215,297,329],"<mml:mo>/</mml:mo>":[120,298],"<mml:mi>O</mml:mi>":[121,248,299],"<mml:mspace":[122,162,165,249,300,332,335],"/>":[123,163,166,250,301,333,336],"<mml:mfenced>":[124,172,181,251,302],"<mml:mo>\u00b7</mml:mo>":[126,304,330],"<mml:msqrt>":[127,305],"</mml:msqrt>":[132,310],"</mml:mfenced>":[133,176,198,254,311],"gates,":[136,314],"have":[138],"$$\\begin{aligned}":[139],"\\mathop":[140],"{{{\\,\\mathrm{\\textbf{Pr}}\\,}}}\\limits":[141],"_{x\\in":[142],"\\left\\{":[143],"0,1\\right\\}":[144],"^n}\\left[":[145],"C(x)=f_k(x)\\right]":[146],"\\leqslant":[147],"\\frac{1}{2}":[148],"+":[149],"\\frac{1}{2^{\\Omega":[150],"(k)}}.":[151],"\\end{aligned}$$":[152],"<mml:mtable>":[156],"<mml:mtr>":[157],"<mml:mtd>":[158],"<mml:munder>":[160],"<mml:mi>Pr</mml:mi>":[164],"<mml:mi>x</mml:mi>":[169,185,195],"<mml:mo>\u2208</mml:mo>":[170],"<mml:mn>0</mml:mn>":[173],"<mml:mo>,</mml:mo>":[174],"<mml:mn>1</mml:mn>":[175,201,206],"</mml:munder>":[180],"<mml:mi>C</mml:mi>":[182],"<mml:mo>\u2a7d</mml:mo>":[199],"<mml:mfrac>":[200,205],"<mml:mn>2</mml:mn>":[202,208,323],"</mml:mfrac>":[203,216],"<mml:mo>+</mml:mo>":[204],"<mml:mi>\u03a9</mml:mi>":[210],"<mml:mo>.</mml:mo>":[217],"</mml:mtd>":[219],"</mml:mtr>":[220],"</mml:mtable>":[221],"This":[224],"lower":[226,231],"bound":[227,232],"matches":[228],"worst-case":[230],"G\u00e1l":[234],"Robere":[236],"by":[237],"letting":[238],"$$k=O\\!\\left(":[239],"n\\right)":[241],".":[257],"$$\\#$$":[258],"<mml:mo>#</mml:mo>":[261],"SAT":[263],"Algorithms":[264],"There":[265],"is":[266],"an":[267],"algorithm":[268],"that":[269],"counts":[270],"number":[272],"satisfying":[274],"assignments":[275],"given":[278],"in":[315],"time":[316],"$$2^{n-k}\\cdot":[317],"{{\\,\\textrm{poly}\\,}}(n)$$":[318],"<mml:mo>-</mml:mo>":[326],"<mml:mtext>poly</mml:mtext>":[334]},"counts_by_year":[],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2022-05-05T00:00:00"}