{"id":"https://openalex.org/W1900995703","doi":"https://doi.org/10.1137/140999220","title":"The Limits of Depth Reduction for Arithmetic Formulas: It's All About the Top Fan-In","display_name":"The Limits of Depth Reduction for Arithmetic Formulas: It's All About the Top Fan-In","publication_year":2015,"publication_date":"2015-01-01","ids":{"openalex":"https://openalex.org/W1900995703","doi":"https://doi.org/10.1137/140999220","mag":"1900995703"},"language":"en","primary_location":{"id":"doi:10.1137/140999220","is_oa":false,"landing_page_url":"https://doi.org/10.1137/140999220","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/A5100713973","display_name":"Mrinal Kumar","orcid":"https://orcid.org/0000-0002-6430-0219"},"institutions":[{"id":"https://openalex.org/I102322142","display_name":"Rutgers, The State University of New Jersey","ror":"https://ror.org/05vt9qd57","country_code":"US","type":"education","lineage":["https://openalex.org/I102322142"]},{"id":"https://openalex.org/I4210096112","display_name":"Rutgers Sexual and Reproductive Health and Rights","ror":"https://ror.org/00rcvgx40","country_code":"NL","type":"other","lineage":["https://openalex.org/I4210096112"]},{"id":"https://openalex.org/I4210123151","display_name":"R\u00fctgers (Germany)","ror":"https://ror.org/02wmkbh90","country_code":"DE","type":"company","lineage":["https://openalex.org/I4210123151"]}],"countries":["DE","NL","US"],"is_corresponding":true,"raw_author_name":"Mrinal Kumar","raw_affiliation_strings":["Rutgers University-New Brunswick"],"affiliations":[{"raw_affiliation_string":"Rutgers University-New Brunswick","institution_ids":["https://openalex.org/I4210123151","https://openalex.org/I102322142","https://openalex.org/I4210096112"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5020540233","display_name":"Shubhangi Saraf","orcid":"https://orcid.org/0009-0005-0874-2978"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Shubhangi Saraf","raw_affiliation_strings":["School of Arts and Sciences, Mathematics"],"affiliations":[{"raw_affiliation_string":"School of Arts and Sciences, Mathematics","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5100713973"],"corresponding_institution_ids":["https://openalex.org/I102322142","https://openalex.org/I4210096112","https://openalex.org/I4210123151"],"apc_list":null,"apc_paid":null,"fwci":1.4021,"has_fulltext":false,"cited_by_count":8,"citation_normalized_percentile":{"value":0.81929616,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"44","issue":"6","first_page":"1601","last_page":"1625"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9994999766349792,"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":0.9994999766349792,"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":0.9958999752998352,"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":0.9907000064849854,"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/mathematics","display_name":"Mathematics","score":0.7384575009346008},{"id":"https://openalex.org/keywords/arithmetic-circuit-complexity","display_name":"Arithmetic circuit complexity","score":0.6847202181816101},{"id":"https://openalex.org/keywords/reduction","display_name":"Reduction (mathematics)","score":0.6013827919960022},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.5712682008743286},{"id":"https://openalex.org/keywords/homogeneous","display_name":"Homogeneous","score":0.5542890429496765},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.5103110671043396},{"id":"https://openalex.org/keywords/electronic-circuit","display_name":"Electronic circuit","score":0.5069170594215393},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.4962332844734192},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4680233895778656},{"id":"https://openalex.org/keywords/circuit-complexity","display_name":"Circuit complexity","score":0.4187326729297638},{"id":"https://openalex.org/keywords/arbitrary-precision-arithmetic","display_name":"Arbitrary-precision arithmetic","score":0.3254014253616333},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.29102495312690735},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.20740777254104614},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.10118839144706726},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.06991586089134216}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7384575009346008},{"id":"https://openalex.org/C155920423","wikidata":"https://www.wikidata.org/wiki/Q4791120","display_name":"Arithmetic circuit complexity","level":3,"score":0.6847202181816101},{"id":"https://openalex.org/C111335779","wikidata":"https://www.wikidata.org/wiki/Q3454686","display_name":"Reduction (mathematics)","level":2,"score":0.6013827919960022},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.5712682008743286},{"id":"https://openalex.org/C66882249","wikidata":"https://www.wikidata.org/wiki/Q169336","display_name":"Homogeneous","level":2,"score":0.5542890429496765},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5103110671043396},{"id":"https://openalex.org/C134146338","wikidata":"https://www.wikidata.org/wiki/Q1815901","display_name":"Electronic circuit","level":2,"score":0.5069170594215393},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.4962332844734192},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4680233895778656},{"id":"https://openalex.org/C90702460","wikidata":"https://www.wikidata.org/wiki/Q1055112","display_name":"Circuit complexity","level":3,"score":0.4187326729297638},{"id":"https://openalex.org/C83581934","wikidata":"https://www.wikidata.org/wiki/Q527381","display_name":"Arbitrary-precision arithmetic","level":2,"score":0.3254014253616333},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.29102495312690735},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.20740777254104614},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.10118839144706726},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.06991586089134216},{"id":"https://openalex.org/C119599485","wikidata":"https://www.wikidata.org/wiki/Q43035","display_name":"Electrical engineering","level":1,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/140999220","is_oa":false,"landing_page_url":"https://doi.org/10.1137/140999220","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":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":7,"referenced_works":["https://openalex.org/W1426742690","https://openalex.org/W2016576580","https://openalex.org/W2026944800","https://openalex.org/W2079910744","https://openalex.org/W2081256023","https://openalex.org/W2093103267","https://openalex.org/W2914183794"],"related_works":["https://openalex.org/W1656092346","https://openalex.org/W2276907117","https://openalex.org/W2181927644","https://openalex.org/W2073476805","https://openalex.org/W2571686129","https://openalex.org/W1559551218","https://openalex.org/W2087454352","https://openalex.org/W2113155657","https://openalex.org/W1584254660","https://openalex.org/W1975269884"],"abstract_inverted_index":{"In":[0,176,291],"recent":[1,113],"years,":[2],"a":[3,94,112,170],"very":[4],"exciting":[5],"and":[6,33,50,53,59,82,100,118,163,183,188,309,312,349,362,372,396],"promising":[7],"method":[8,20,23,391],"for":[9,13,104,131,140,158,192,216,240,251,265,268,322,347,410],"proving":[10],"lower":[11,102,129,156,214,263,408],"bounds":[12,103,130,157,215,264,409],"arithmetic":[14,108,133,161,173,193,206,413],"circuits":[15,223,246,267],"has":[16,247],"been":[17],"proposed.":[18],"This":[19,243],"combines":[21],"the":[22,29,60,63,71,181,200,211,217,252,260,285,296,327,353,390],"of":[24,31,62,73,96,107,115,185,202,219,231,245,254,300,329,352,357,392],"depth":[25,138,186,203,221,238,279,297,343,394],"reduction":[26,139,187,239,280,298,344,395],"developed":[27,69],"in":[28,70,110,172,386],"works":[30,72],"Agrawal":[32],"Vinay":[34],"[FOCS,":[35],"IEEE,":[36],"Piscataway,":[37],"NJ,":[38],"2008,":[39],"pp.":[40,48,125,307,370],"67--75],":[41],"Koiran":[42,301],"[Theoret.":[43,302],"Comput.":[44,303],"Sci.,":[45,304],"448":[46,305],"(2012),":[47,306,315],"56--65],":[49],"Tavenas":[51,310],"[Inform.":[52,311],"Comput.,":[54,313,335,379],"240":[55,314],"(2015),":[56,337,381],"pp,":[57,316,338,382],"2--11],":[58],"use":[61],"shifted":[64,189,397],"partial":[65,190,398],"derivative":[66],"complexity":[67],"measure":[68],"Kayal":[74],"[Electronic":[75],"Colloquium":[76],"on":[77],"Computational":[78],"Complexity,":[79],"TR12-081,":[80],"2012]":[81],"Gupta":[83],"et":[84,331,375],"al.":[85,332,376],"[J.":[86],"ACM,":[87,121,366],"61":[88],"(2014),":[89],"33].":[90],"These":[91],"results":[92,99,328,385],"inspired":[93],"flurry":[95],"other":[97],"beautiful":[98],"strong":[101],"various":[105],"classes":[106],"circuits,":[109,348],"particular":[111,292,387],"work":[114],"Kayal,":[116,359],"Saha,":[117,361],"Saptharishi":[119],"[STOC,":[120],"New":[122,367],"York,":[123,368],"2014,":[124,369],"146--153]":[126,371],"showing":[127],"superpolynomial":[128,155,213,407],"regular":[132],"formulas":[134],"via":[135,198],"an":[136,147],"improved":[137,237,278,320,393],"these":[141,151,241,266],"formulas.":[142,194,414],"It":[143],"was":[144],"left":[145],"as":[146],"intriguing":[148],"question":[149],"if":[150,164],"methods":[152],"could":[153],"prove":[154,406],"general":[159],"(homogeneous)":[160],"formulas,":[162,324],"so":[165],"this":[166,177,293],"would":[167],"indeed":[168],"be":[169,319,402],"breakthrough":[171],"circuit":[174],"complexity.":[175],"paper":[178],"we":[179],"study":[180],"power":[182],"limitations":[184],"derivatives":[191,399],"We":[195,208,258,275],"do":[196],"it":[197],"studying":[199],"class":[201,218,244],"4":[204,222],"homogeneous":[205,220,323],"circuits.":[207,242],"show:":[209],"(1)":[210],"first":[212,261],"with":[224],"top":[225,270,286],"fan-in":[226,271,287],"$o(\\log":[227],"n)$.":[228,290],"The":[229],"core":[230],"our":[232],"result":[233],"is":[234,281,288,345],"to":[235,405],"show":[236,276],"received":[248],"much":[249],"attention":[250],"problem":[253],"polynomial":[255],"identity":[256],"testing.":[257],"give":[259],"nontrivial":[262],"any":[269],"$\\geq":[272],"2$.":[273],"(2)":[274],"that":[277,295,342,389],"not":[282,401],"possible":[283],"when":[284],"$\\Omega(\\log":[289],"shows":[294],"procedure":[299],"56--65]":[308],"2--11]":[317],"cannot":[318],"even":[321],"thus":[325],"strengthening":[326],"Fournier":[330,374],"[SIAM":[333],"J.":[334,378],"44":[336,380],"1173--1201]":[339],"who":[340],"showed":[341],"tight":[346],"answering":[350],"some":[351],"main":[354],"open":[355],"questions":[356],"[N.":[358],"C.":[360],"R.":[363],"Saptharishi,":[364],"STOC,":[365],"[H.":[373],"SIAM":[377],"1173--1201].":[383],"Our":[384],"suggest":[388],"may":[400],"powerful":[403],"enough":[404],"(even":[411],"homogeneous)":[412]},"counts_by_year":[{"year":2024,"cited_by_count":2},{"year":2021,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":3}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
