{"id":"https://openalex.org/W2003519998","doi":"https://doi.org/10.1137/0215070","title":"Log Depth Circuits for Division and Related Problems","display_name":"Log Depth Circuits for Division and Related Problems","publication_year":1986,"publication_date":"1986-11-01","ids":{"openalex":"https://openalex.org/W2003519998","doi":"https://doi.org/10.1137/0215070","mag":"2003519998"},"language":"en","primary_location":{"id":"doi:10.1137/0215070","is_oa":false,"landing_page_url":"https://doi.org/10.1137/0215070","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/A5042076941","display_name":"Paul Beame","orcid":"https://orcid.org/0000-0002-2666-3545"},"institutions":[{"id":"https://openalex.org/I185261750","display_name":"University of Toronto","ror":"https://ror.org/03dbr7087","country_code":"CA","type":"education","lineage":["https://openalex.org/I185261750"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Paul W. Beame","raw_affiliation_strings":["University of Toronto"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Toronto","institution_ids":["https://openalex.org/I185261750"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5034555443","display_name":"Stephen Cook","orcid":null},"institutions":[{"id":"https://openalex.org/I185261750","display_name":"University of Toronto","ror":"https://ror.org/03dbr7087","country_code":"CA","type":"education","lineage":["https://openalex.org/I185261750"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Stephen A. Cook","raw_affiliation_strings":["University of Toronto"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Toronto","institution_ids":["https://openalex.org/I185261750"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5066858573","display_name":"H. James Hoover","orcid":null},"institutions":[{"id":"https://openalex.org/I185261750","display_name":"University of Toronto","ror":"https://ror.org/03dbr7087","country_code":"CA","type":"education","lineage":["https://openalex.org/I185261750"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"H. James Hoover","raw_affiliation_strings":["University of Toronto"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Toronto","institution_ids":["https://openalex.org/I185261750"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":11.379,"has_fulltext":false,"cited_by_count":238,"citation_normalized_percentile":{"value":0.98978897,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":91,"max":99},"biblio":{"volume":"15","issue":"4","first_page":"994","last_page":"1003"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11697","display_name":"Numerical Methods and Algorithms","score":0.9984999895095825,"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/T11697","display_name":"Numerical Methods and Algorithms","score":0.9984999895095825,"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/T11522","display_name":"VLSI and FPGA Design Techniques","score":0.9973999857902527,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11032","display_name":"VLSI and Analog Circuit Testing","score":0.9966999888420105,"subfield":{"id":"https://openalex.org/subfields/1708","display_name":"Hardware and Architecture"},"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/divisibility-rule","display_name":"Divisibility rule","score":0.7736233472824097},{"id":"https://openalex.org/keywords/division","display_name":"Division (mathematics)","score":0.759926438331604},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6969041228294373},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.5972344875335693},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.5723579525947571},{"id":"https://openalex.org/keywords/boolean-circuit","display_name":"Boolean circuit","score":0.5576827526092529},{"id":"https://openalex.org/keywords/circuit-complexity","display_name":"Circuit complexity","score":0.5418132543563843},{"id":"https://openalex.org/keywords/electronic-circuit","display_name":"Electronic circuit","score":0.5132369995117188},{"id":"https://openalex.org/keywords/boolean-function","display_name":"Boolean function","score":0.48868587613105774},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.45278167724609375},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4029942750930786},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.36305609345436096},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.2803715765476227},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.25354433059692383}],"concepts":[{"id":"https://openalex.org/C9991821","wikidata":"https://www.wikidata.org/wiki/Q858520","display_name":"Divisibility rule","level":2,"score":0.7736233472824097},{"id":"https://openalex.org/C60798267","wikidata":"https://www.wikidata.org/wiki/Q1226939","display_name":"Division (mathematics)","level":2,"score":0.759926438331604},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6969041228294373},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.5972344875335693},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.5723579525947571},{"id":"https://openalex.org/C141796577","wikidata":"https://www.wikidata.org/wiki/Q837479","display_name":"Boolean circuit","level":3,"score":0.5576827526092529},{"id":"https://openalex.org/C90702460","wikidata":"https://www.wikidata.org/wiki/Q1055112","display_name":"Circuit complexity","level":3,"score":0.5418132543563843},{"id":"https://openalex.org/C134146338","wikidata":"https://www.wikidata.org/wiki/Q1815901","display_name":"Electronic circuit","level":2,"score":0.5132369995117188},{"id":"https://openalex.org/C187455244","wikidata":"https://www.wikidata.org/wiki/Q942353","display_name":"Boolean function","level":2,"score":0.48868587613105774},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.45278167724609375},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4029942750930786},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.36305609345436096},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.2803715765476227},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.25354433059692383},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","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},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C119599485","wikidata":"https://www.wikidata.org/wiki/Q43035","display_name":"Electrical engineering","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/0215070","is_oa":false,"landing_page_url":"https://doi.org/10.1137/0215070","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:CiteSeerX.psu:10.1.1.698.9492","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.698.9492","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://homes.cs.washington.edu/%7Ebeame/papers/division.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":5,"referenced_works":["https://openalex.org/W1558316751","https://openalex.org/W2027501230","https://openalex.org/W2088141885","https://openalex.org/W2126466445","https://openalex.org/W4235903219"],"related_works":["https://openalex.org/W2009341260","https://openalex.org/W3204916073","https://openalex.org/W2952185983","https://openalex.org/W2055594799","https://openalex.org/W2403507963","https://openalex.org/W2611822924","https://openalex.org/W2486254595","https://openalex.org/W2799890067","https://openalex.org/W1492165320","https://openalex.org/W3046459465"],"abstract_inverted_index":{"We":[0,16],"present":[1],"optimal":[2,42],"depth":[3,27,45],"Boolean":[4],"circuits":[5],"(depth":[6],"$O(\\log":[7],"n)$)":[8],"for":[9,37,43],"integer":[10],"division,":[11],"powering,":[12],"and":[13,28,46],"multiple":[14],"products.":[15],"also":[17],"show":[18],"that":[19,40],"these":[20],"three":[21],"problems":[22],"are":[23],"of":[24],"equivalent":[25],"uniform":[26],"space":[29],"complexity.":[30],"In":[31],"addition,":[32],"we":[33],"describe":[34],"an":[35],"algorithm":[36],"testing":[38],"divisibility":[39],"is":[41],"both":[44],"space.":[47]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":4},{"year":2022,"cited_by_count":8},{"year":2021,"cited_by_count":9},{"year":2020,"cited_by_count":4},{"year":2019,"cited_by_count":9},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":6},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":4},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":7},{"year":2012,"cited_by_count":3}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}