{"id":"https://openalex.org/W3215601401","doi":"https://doi.org/10.1007/s00454-023-00584-z","title":"Computing Generalized Rank Invariant for 2-Parameter Persistence Modules via Zigzag Persistence and Its Applications","display_name":"Computing Generalized Rank Invariant for 2-Parameter Persistence Modules via Zigzag Persistence and Its Applications","publication_year":2023,"publication_date":"2023-10-07","ids":{"openalex":"https://openalex.org/W3215601401","doi":"https://doi.org/10.1007/s00454-023-00584-z","mag":"3215601401"},"language":"en","primary_location":{"id":"doi:10.1007/s00454-023-00584-z","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00454-023-00584-z","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00454-023-00584-z.pdf","source":{"id":"https://openalex.org/S103671469","display_name":"Discrete & Computational Geometry","issn_l":"0179-5376","issn":["0179-5376","1432-0444"],"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":"Discrete &amp; Computational Geometry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s00454-023-00584-z.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5025354167","display_name":"Tamal K. Dey","orcid":"https://orcid.org/0000-0001-5160-9738"},"institutions":[{"id":"https://openalex.org/I219193219","display_name":"Purdue University West Lafayette","ror":"https://ror.org/02dqehb95","country_code":"US","type":"education","lineage":["https://openalex.org/I219193219"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Tamal K. Dey","raw_affiliation_strings":["Department of Computer Science, Purdue University, West Lafayette, IN, USA"],"raw_orcid":"https://orcid.org/0000-0001-5160-9738","affiliations":[{"raw_affiliation_string":"Department of Computer Science, Purdue University, West Lafayette, IN, USA","institution_ids":["https://openalex.org/I219193219"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100386550","display_name":"Woojin Kim","orcid":"https://orcid.org/0000-0001-8081-5872"},"institutions":[{"id":"https://openalex.org/I157485424","display_name":"Korea Advanced Institute of Science and Technology","ror":"https://ror.org/05apxxy63","country_code":"KR","type":"education","lineage":["https://openalex.org/I157485424"]},{"id":"https://openalex.org/I4210158432","display_name":"National Institute for Mathematical Sciences","ror":"https://ror.org/04n7py080","country_code":"KR","type":"education","lineage":["https://openalex.org/I4210158432"]}],"countries":["KR"],"is_corresponding":true,"raw_author_name":"Woojin Kim","raw_affiliation_strings":["Department of Mathematical Sciences, KAIST, Daejeon, South Korea"],"raw_orcid":"https://orcid.org/0000-0001-8081-5872","affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, KAIST, Daejeon, South Korea","institution_ids":["https://openalex.org/I4210158432","https://openalex.org/I157485424"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5016256957","display_name":"Facundo M\u00e9moli","orcid":"https://orcid.org/0000-0001-8409-0549"},"institutions":[{"id":"https://openalex.org/I52357470","display_name":"The Ohio State University","ror":"https://ror.org/00rs6vg23","country_code":"US","type":"education","lineage":["https://openalex.org/I52357470"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Facundo M\u00e9moli","raw_affiliation_strings":["Department of Mathematics and Department of Computer Science and Engineering, The Ohio State University, Columbus, OH, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Department of Computer Science and Engineering, The Ohio State University, Columbus, OH, USA","institution_ids":["https://openalex.org/I52357470"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5100386550"],"corresponding_institution_ids":["https://openalex.org/I157485424","https://openalex.org/I4210158432"],"apc_list":{"value":2290,"currency":"EUR","value_usd":2890},"apc_paid":{"value":2290,"currency":"EUR","value_usd":2890},"fwci":0.3822,"has_fulltext":true,"cited_by_count":3,"citation_normalized_percentile":{"value":0.60726546,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":95},"biblio":{"volume":"71","issue":"1","first_page":"67","last_page":"94"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12536","display_name":"Topological and Geometric Data Analysis","score":0.9998999834060669,"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/T12536","display_name":"Topological and Geometric Data Analysis","score":0.9998999834060669,"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/T10896","display_name":"Homotopy and Cohomology in Algebraic Topology","score":0.9814000129699707,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11304","display_name":"Advanced Neuroimaging Techniques and Applications","score":0.9735000133514404,"subfield":{"id":"https://openalex.org/subfields/2741","display_name":"Radiology, Nuclear Medicine and Imaging"},"field":{"id":"https://openalex.org/fields/27","display_name":"Medicine"},"domain":{"id":"https://openalex.org/domains/4","display_name":"Health Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.5555126667022705},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.38164621591567993},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.3625386953353882}],"concepts":[{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.5555126667022705},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.38164621591567993},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3625386953353882}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s00454-023-00584-z","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00454-023-00584-z","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00454-023-00584-z.pdf","source":{"id":"https://openalex.org/S103671469","display_name":"Discrete & Computational Geometry","issn_l":"0179-5376","issn":["0179-5376","1432-0444"],"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":"Discrete &amp; Computational Geometry","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1007/s00454-023-00584-z","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00454-023-00584-z","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00454-023-00584-z.pdf","source":{"id":"https://openalex.org/S103671469","display_name":"Discrete & Computational Geometry","issn_l":"0179-5376","issn":["0179-5376","1432-0444"],"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":"Discrete &amp; Computational Geometry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2137412393","display_name":null,"funder_award_id":"DMS-1547357","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G2351444152","display_name":null,"funder_award_id":"CCF-2049010","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G3231818491","display_name":"RI: Medium:Collaborative Research: Through synapses to spatial learning: a topological approach","funder_award_id":"1901360","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G4229663209","display_name":null,"funder_award_id":"1547357","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G5246179914","display_name":null,"funder_award_id":"CCF-1740761","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6073240786","display_name":null,"funder_award_id":"IIS-1901360","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6367296400","display_name":null,"funder_award_id":"1740761","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":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3215601401.pdf"},"referenced_works_count":41,"referenced_works":["https://openalex.org/W1496383623","https://openalex.org/W1516374480","https://openalex.org/W1550890970","https://openalex.org/W1896687453","https://openalex.org/W1964172127","https://openalex.org/W1993818927","https://openalex.org/W2040661328","https://openalex.org/W2069314032","https://openalex.org/W2098825762","https://openalex.org/W2110369778","https://openalex.org/W2112471150","https://openalex.org/W2120713972","https://openalex.org/W2154477220","https://openalex.org/W2185147444","https://openalex.org/W2335810221","https://openalex.org/W2588688245","https://openalex.org/W2903721645","https://openalex.org/W2913946465","https://openalex.org/W2918466505","https://openalex.org/W2928401208","https://openalex.org/W2936375853","https://openalex.org/W2942078528","https://openalex.org/W2964292054","https://openalex.org/W2969632132","https://openalex.org/W2990798340","https://openalex.org/W2991962490","https://openalex.org/W2995087345","https://openalex.org/W3007184001","https://openalex.org/W3011959994","https://openalex.org/W3037433535","https://openalex.org/W3092727470","https://openalex.org/W3116783414","https://openalex.org/W3140579943","https://openalex.org/W3179551217","https://openalex.org/W3185924222","https://openalex.org/W3211227960","https://openalex.org/W4213033583","https://openalex.org/W4251980118","https://openalex.org/W4285719527","https://openalex.org/W6600721425","https://openalex.org/W6826254890"],"related_works":["https://openalex.org/W1979597421","https://openalex.org/W2007980826","https://openalex.org/W4245490552","https://openalex.org/W4225152035","https://openalex.org/W2061531152","https://openalex.org/W3002753104","https://openalex.org/W2077600819","https://openalex.org/W1587224694","https://openalex.org/W2911598644","https://openalex.org/W2042127053"],"abstract_inverted_index":{"Abstract":[0],"The":[1],"notion":[2],"of":[3,9,50,70,94,102,117,121,135,179],"generalized":[4,24,43,68,92],"rank":[5,44,69,93],"in":[6,36,81],"the":[7,37,42,67,71,91,100,103,115,177,192],"context":[8],"multiparameter":[10],"persistence":[11,25,61],"has":[12,31],"become":[13],"an":[14,188],"important":[15],"ingredient":[16],"for":[17,130,191],"defining":[18],"interesting":[19],"homological":[20],"structures":[21],"such":[22],"as":[23],"diagrams.":[26],"However,":[27],"its":[28,85,216],"efficient":[29],"computation":[30,143],"not":[32],"yet":[33],"been":[34],"studied":[35],"literature.":[38],"We":[39,182],"show":[40],"that":[41,74,110],"over":[45,96],"a":[46,51,78,118,196,199],"finite":[47],"interval":[48,207],"I":[49,82,97,133],"$$\\textbf{Z}^2$$":[52],"<mml:math":[53,123,137,147,164],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">":[54,124,138,148,165],"<mml:msup>":[55,152],"<mml:mi>Z</mml:mi>":[56],"<mml:mn>2</mml:mn>":[57,170],"</mml:msup>":[58,155],"</mml:math>":[59,126,140,158,175],"-indexed":[60],"module":[62,73,105,200],"M":[63,95,113,201,205],"is":[64,75,114,176,206],"equal":[65],"to":[66,109,186],"zigzag":[72,104],"induced":[76],"on":[77],"certain":[79],"path":[80],"tracing":[83],"mostly":[84],"boundary.":[86],"Hence,":[87],"we":[88],"can":[89],"compute":[90,212],"by":[98,107],"computing":[99],"barcode":[101],"obtained":[106],"restricting":[108],"path.":[111],"If":[112],"homology":[116],"bifiltration":[119,197],"F":[120],"$$t$$":[122,136],"<mml:mi>t</mml:mi>":[125,139,153],"simplices":[127],"(while":[128],"accounting":[129],"multi-criticality)":[131],"and":[132],"consists":[134],"points,":[141],"this":[142,184],"takes":[144],"$$O(t^\\omega":[145],")$$":[146],"<mml:mrow>":[149,166],"<mml:mi>O</mml:mi>":[150],"<mml:mo>(</mml:mo>":[151],"<mml:mi>\u03c9</mml:mi>":[154,167],"<mml:mo>)</mml:mo>":[156,173],"</mml:mrow>":[157,174],"time":[159],"where":[160],"$$\\omega":[161],"\\in":[162],"[2,2.373)$$":[163],"<mml:mo>\u2208</mml:mo>":[168],"<mml:mo>[</mml:mo>":[169],"<mml:mo>,</mml:mo>":[171],"<mml:mn>2.373</mml:mn>":[172],"exponent":[178],"matrix":[180],"multiplication.":[181],"apply":[183],"result":[185],"obtain":[187],"improved":[189],"algorithm":[190],"following":[193],"problem.":[194],"Given":[195],"inducing":[198],",":[202],"determine":[203],"whether":[204],"decomposable":[208],"and,":[209],"if":[210],"so,":[211],"all":[213],"intervals":[214],"supporting":[215],"indecomposable":[217],"summands.":[218]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1}],"updated_date":"2026-05-16T08:24:45.110214","created_date":"2025-10-10T00:00:00"}
