{"id":"https://openalex.org/W2125658617","doi":"https://doi.org/10.1142/s0218195994000069","title":"EFFICIENT ENUMERATION OF GRID POINTS IN A CONVEX POLYGON AND ITS APPLICATION TO INTEGER PROGRAMMING","display_name":"EFFICIENT ENUMERATION OF GRID POINTS IN A CONVEX POLYGON AND ITS APPLICATION TO INTEGER PROGRAMMING","publication_year":1994,"publication_date":"1994-03-01","ids":{"openalex":"https://openalex.org/W2125658617","doi":"https://doi.org/10.1142/s0218195994000069","mag":"2125658617"},"language":"en","primary_location":{"id":"doi:10.1142/s0218195994000069","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218195994000069","pdf_url":null,"source":{"id":"https://openalex.org/S25827807","display_name":"International Journal of Computational Geometry & Applications","issn_l":"0218-1959","issn":["0218-1959","1793-6357"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Computational Geometry & Applications","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/A5047592572","display_name":"Naoyoshi Kanamaru","orcid":null},"institutions":[{"id":"https://openalex.org/I201537933","display_name":"Tohoku University","ror":"https://ror.org/01dq60k83","country_code":"JP","type":"education","lineage":["https://openalex.org/I201537933"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"NAOYOSHI KANAMARU","raw_affiliation_strings":["Faculty of Engineering, Tohoku University, Sendai 980, Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Faculty of Engineering, Tohoku University, Sendai 980, Japan","institution_ids":["https://openalex.org/I201537933"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5033580185","display_name":"Takao Nishizeki","orcid":null},"institutions":[{"id":"https://openalex.org/I201537933","display_name":"Tohoku University","ror":"https://ror.org/01dq60k83","country_code":"JP","type":"education","lineage":["https://openalex.org/I201537933"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"TAKAO NISHIZEKI","raw_affiliation_strings":["Faculty of Engineering, Tohoku University, Sendai 980, Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Faculty of Engineering, Tohoku University, Sendai 980, Japan","institution_ids":["https://openalex.org/I201537933"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5038413661","display_name":"Tetsuo Asano","orcid":"https://orcid.org/0000-0002-4741-0467"},"institutions":[{"id":"https://openalex.org/I189513530","display_name":"Osaka Electro-Communication University","ror":"https://ror.org/056bksm23","country_code":"JP","type":"education","lineage":["https://openalex.org/I189513530"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"TETSUO ASANO","raw_affiliation_strings":["Osaka Electro-Communication University, Neyagawa 572, Japan","Osaka Electro-Communication University Neyagawa, 572, Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Osaka Electro-Communication University, Neyagawa 572, Japan","institution_ids":["https://openalex.org/I189513530"]},{"raw_affiliation_string":"Osaka Electro-Communication University Neyagawa, 572, Japan","institution_ids":["https://openalex.org/I189513530"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.3894,"has_fulltext":false,"cited_by_count":9,"citation_normalized_percentile":{"value":0.64689064,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"04","issue":"01","first_page":"69","last_page":"85"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9972000122070312,"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"}},"topics":[{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9972000122070312,"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/T12176","display_name":"Optimization and Packing Problems","score":0.9945999979972839,"subfield":{"id":"https://openalex.org/subfields/2209","display_name":"Industrial and Manufacturing 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/T11106","display_name":"Data Management and Algorithms","score":0.9817000031471252,"subfield":{"id":"https://openalex.org/subfields/1711","display_name":"Signal Processing"},"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":0.753516435623169},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6559560894966125},{"id":"https://openalex.org/keywords/rectangle","display_name":"Rectangle","score":0.6013669967651367},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.5989239811897278},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.596755862236023},{"id":"https://openalex.org/keywords/polygon","display_name":"Polygon (computer graphics)","score":0.594433069229126},{"id":"https://openalex.org/keywords/grid","display_name":"Grid","score":0.5903286337852478},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.5789778232574463},{"id":"https://openalex.org/keywords/convex-polygon","display_name":"Convex polygon","score":0.5065988302230835},{"id":"https://openalex.org/keywords/simple","display_name":"Simple (philosophy)","score":0.50501948595047},{"id":"https://openalex.org/keywords/integer-points-in-convex-polyhedra","display_name":"Integer points in convex polyhedra","score":0.499192476272583},{"id":"https://openalex.org/keywords/enumeration","display_name":"Enumeration","score":0.4649077355861664},{"id":"https://openalex.org/keywords/integer-programming","display_name":"Integer programming","score":0.4592326879501343},{"id":"https://openalex.org/keywords/polygon-covering","display_name":"Polygon covering","score":0.4236277937889099},{"id":"https://openalex.org/keywords/time-complexity","display_name":"Time complexity","score":0.42289745807647705},{"id":"https://openalex.org/keywords/simple-polygon","display_name":"Simple polygon","score":0.4148111045360565},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.34547159075737},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.27800118923187256},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.1638699173927307},{"id":"https://openalex.org/keywords/branch-and-price","display_name":"Branch and price","score":0.1017855703830719},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.09619209170341492}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.753516435623169},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6559560894966125},{"id":"https://openalex.org/C2781302577","wikidata":"https://www.wikidata.org/wiki/Q209","display_name":"Rectangle","level":2,"score":0.6013669967651367},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.5989239811897278},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.596755862236023},{"id":"https://openalex.org/C190694206","wikidata":"https://www.wikidata.org/wiki/Q3276654","display_name":"Polygon (computer graphics)","level":3,"score":0.594433069229126},{"id":"https://openalex.org/C187691185","wikidata":"https://www.wikidata.org/wiki/Q2020720","display_name":"Grid","level":2,"score":0.5903286337852478},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.5789778232574463},{"id":"https://openalex.org/C2780034980","wikidata":"https://www.wikidata.org/wiki/Q3259505","display_name":"Convex polygon","level":3,"score":0.5065988302230835},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.50501948595047},{"id":"https://openalex.org/C147700949","wikidata":"https://www.wikidata.org/wiki/Q6042594","display_name":"Integer points in convex polyhedra","level":4,"score":0.499192476272583},{"id":"https://openalex.org/C156340839","wikidata":"https://www.wikidata.org/wiki/Q2704791","display_name":"Enumeration","level":2,"score":0.4649077355861664},{"id":"https://openalex.org/C56086750","wikidata":"https://www.wikidata.org/wiki/Q6042592","display_name":"Integer programming","level":2,"score":0.4592326879501343},{"id":"https://openalex.org/C187778580","wikidata":"https://www.wikidata.org/wiki/Q18389289","display_name":"Polygon covering","level":3,"score":0.4236277937889099},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.42289745807647705},{"id":"https://openalex.org/C197949415","wikidata":"https://www.wikidata.org/wiki/Q782746","display_name":"Simple polygon","level":3,"score":0.4148111045360565},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.34547159075737},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.27800118923187256},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.1638699173927307},{"id":"https://openalex.org/C123558587","wikidata":"https://www.wikidata.org/wiki/Q4956382","display_name":"Branch and price","level":3,"score":0.1017855703830719},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.09619209170341492},{"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/C126042441","wikidata":"https://www.wikidata.org/wiki/Q1324888","display_name":"Frame (networking)","level":2,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0},{"id":"https://openalex.org/C76155785","wikidata":"https://www.wikidata.org/wiki/Q418","display_name":"Telecommunications","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1142/s0218195994000069","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218195994000069","pdf_url":null,"source":{"id":"https://openalex.org/S25827807","display_name":"International Journal of Computational Geometry & Applications","issn_l":"0218-1959","issn":["0218-1959","1793-6357"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Computational Geometry & Applications","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":0,"referenced_works":[],"related_works":["https://openalex.org/W2038184042","https://openalex.org/W2997571660","https://openalex.org/W4288091041","https://openalex.org/W2980991945","https://openalex.org/W2165083794","https://openalex.org/W3036994771","https://openalex.org/W1579553590","https://openalex.org/W2016736910","https://openalex.org/W2004395316","https://openalex.org/W2913852997"],"abstract_inverted_index":{"This":[0,99],"paper":[1,60,110],"first":[2],"presents":[3,112],"an":[4,53],"algorithm":[5,65,104],"for":[6,114],"enumerating":[7],"all":[8],"the":[9,28,37,40,43,47,57,87,95,101,116],"integer-grid":[10],"points":[11,33,120],"in":[12,17,76,105,121],"a":[13,63,68,90,122,125],"given":[14],"convex":[15,91],"m-gon":[16],"O(K":[18],"+":[19,21,80],"m":[20,74,79],"log":[22,78,81],"n)":[23,82],"time":[24,83],"where":[25,84],"K":[26],"is":[27,36,86],"number":[29,117],"of":[30,39,46,52,89,118],"such":[31],"grid":[32,119],"and":[34,49,107],"n":[35,85],"dimension":[38,88],"m-gon,":[41],"i.e.,":[42],"shorter":[44],"length":[45],"horizontal":[48],"vertical":[50],"sides":[51],"axis-parallel":[54],"rectangle":[55],"enclosing":[56],"m-gon.":[58],"The":[59,109],"next":[61],"gives":[62],"simple":[64,126],"which":[66],"solves":[67],"two-variable":[69],"integer":[70],"programming":[71],"problem":[72],"with":[73],"constraints":[75],"O(m":[77],"polygon":[92],"corresponding":[93],"to":[94],"feasible":[96],"solution":[97],"space.":[98],"improves":[100],"best":[102],"known":[103],"complexity":[106],"simplicity.":[108],"finally":[111],"algorithms":[113],"counting":[115],"triangle":[123],"or":[124],"polygon.":[127]},"counts_by_year":[],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}