{"id":"https://openalex.org/W4378193952","doi":"https://doi.org/10.1007/s00454-023-00492-2","title":"Unbounded Regions of High-Order Voronoi Diagrams of Lines and Line Segments in Higher Dimensions","display_name":"Unbounded Regions of High-Order Voronoi Diagrams of Lines and Line Segments in Higher Dimensions","publication_year":2023,"publication_date":"2023-05-25","ids":{"openalex":"https://openalex.org/W4378193952","doi":"https://doi.org/10.1007/s00454-023-00492-2","pmid":"https://pubmed.ncbi.nlm.nih.gov/39376993"},"language":"en","primary_location":{"id":"doi:10.1007/s00454-023-00492-2","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00454-023-00492-2","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00454-023-00492-2.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","pubmed"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s00454-023-00492-2.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5059958789","display_name":"Gill Barequet","orcid":"https://orcid.org/0000-0001-7501-6240"},"institutions":[{"id":"https://openalex.org/I174306211","display_name":"Technion \u2013 Israel Institute of Technology","ror":"https://ror.org/03qryx823","country_code":"IL","type":"education","lineage":["https://openalex.org/I174306211"]}],"countries":["IL"],"is_corresponding":false,"raw_author_name":"Gill Barequet","raw_affiliation_strings":["Dept. of Computer Science, The Technion - Israel Inst. of Technology, Haifa, 3200003 Israel","Dept. of Computer Science, The Technion \u2013 Israel Inst. of Technology, Haifa, 3200003, Israel"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Dept. of Computer Science, The Technion - Israel Inst. of Technology, Haifa, 3200003 Israel","institution_ids":["https://openalex.org/I174306211"]},{"raw_affiliation_string":"Dept. of Computer Science, The Technion \u2013 Israel Inst. of Technology, Haifa, 3200003, Israel","institution_ids":["https://openalex.org/I174306211"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5046472103","display_name":"Evanthia Papadopoulou","orcid":"https://orcid.org/0000-0003-0144-7384"},"institutions":[{"id":"https://openalex.org/I57201433","display_name":"Universit\u00e0 della Svizzera italiana","ror":"https://ror.org/03c4atk17","country_code":"CH","type":"education","lineage":["https://openalex.org/I57201433"]}],"countries":["CH"],"is_corresponding":true,"raw_author_name":"Evanthia Papadopoulou","raw_affiliation_strings":["Faculty of Informatics, Universit\u00e0 della Svizzera italiana, 6900 Lugano, Switzerland","Faculty of Informatics, Universit\u00e0 della Svizzera italiana, 6900, Lugano, Switzerland"],"raw_orcid":"https://orcid.org/0000-0003-0144-7384","affiliations":[{"raw_affiliation_string":"Faculty of Informatics, Universit\u00e0 della Svizzera italiana, 6900 Lugano, Switzerland","institution_ids":["https://openalex.org/I57201433"]},{"raw_affiliation_string":"Faculty of Informatics, Universit\u00e0 della Svizzera italiana, 6900, Lugano, Switzerland","institution_ids":["https://openalex.org/I57201433"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5046430864","display_name":"Martin Suderland","orcid":"https://orcid.org/0000-0002-6604-6381"},"institutions":[{"id":"https://openalex.org/I57201433","display_name":"Universit\u00e0 della Svizzera italiana","ror":"https://ror.org/03c4atk17","country_code":"CH","type":"education","lineage":["https://openalex.org/I57201433"]}],"countries":["CH"],"is_corresponding":false,"raw_author_name":"Martin Suderland","raw_affiliation_strings":["Faculty of Informatics, Universit\u00e0 della Svizzera italiana, 6900 Lugano, Switzerland","Faculty of Informatics, Universit\u00e0 della Svizzera italiana, 6900, Lugano, Switzerland"],"raw_orcid":"https://orcid.org/0000-0002-6604-6381","affiliations":[{"raw_affiliation_string":"Faculty of Informatics, Universit\u00e0 della Svizzera italiana, 6900 Lugano, Switzerland","institution_ids":["https://openalex.org/I57201433"]},{"raw_affiliation_string":"Faculty of Informatics, Universit\u00e0 della Svizzera italiana, 6900, Lugano, Switzerland","institution_ids":["https://openalex.org/I57201433"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5046472103"],"corresponding_institution_ids":["https://openalex.org/I57201433"],"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":0.10163791,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"72","issue":"3","first_page":"1304","last_page":"1332"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":1.0,"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":1.0,"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/T11164","display_name":"Remote Sensing and LiDAR Applications","score":0.9991000294685364,"subfield":{"id":"https://openalex.org/subfields/2305","display_name":"Environmental Engineering"},"field":{"id":"https://openalex.org/fields/23","display_name":"Environmental Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12923","display_name":"Digital Image Processing Techniques","score":0.9983999729156494,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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/algorithm","display_name":"Algorithm","score":0.5269879698753357},{"id":"https://openalex.org/keywords/voronoi-diagram","display_name":"Voronoi diagram","score":0.42687633633613586},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.2764262557029724},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.18538153171539307},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.1104578971862793}],"concepts":[{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.5269879698753357},{"id":"https://openalex.org/C24881265","wikidata":"https://www.wikidata.org/wiki/Q757267","display_name":"Voronoi diagram","level":2,"score":0.42687633633613586},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2764262557029724},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.18538153171539307},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.1104578971862793}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1007/s00454-023-00492-2","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00454-023-00492-2","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00454-023-00492-2.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"},{"id":"pmid:39376993","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/39376993","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Discrete & computational geometry","raw_type":null},{"id":"pmh:oai:pubmedcentral.nih.gov:11455698","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/11455698","pdf_url":"https://pmc.ncbi.nlm.nih.gov/articles/PMC11455698/pdf/454_2023_Article_492.pdf","source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Discrete Comput Geom","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.1007/s00454-023-00492-2","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00454-023-00492-2","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00454-023-00492-2.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/G1845367639","display_name":"Dimension++","funder_award_id":"201356","funder_id":"https://openalex.org/F4320320924","funder_display_name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung"},{"id":"https://openalex.org/G8090643491","display_name":null,"funder_award_id":"2017684","funder_id":"https://openalex.org/F4320308960","funder_display_name":"United States - Israel Binational Science Foundation"},{"id":"https://openalex.org/G8617913352","display_name":null,"funder_award_id":"200021E-201356","funder_id":"https://openalex.org/F4320320924","funder_display_name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320308960","display_name":"United States - Israel Binational Science Foundation","ror":"https://ror.org/00j8z2m73"},{"id":"https://openalex.org/F4320320924","display_name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung","ror":"https://ror.org/00yjd3n13"}],"has_content":{"pdf":true,"grobid_xml":false},"content_urls":{"pdf":"https://content.openalex.org/works/W4378193952.pdf"},"referenced_works_count":28,"referenced_works":["https://openalex.org/W1592877305","https://openalex.org/W1973073235","https://openalex.org/W1982041519","https://openalex.org/W1984529618","https://openalex.org/W1986837908","https://openalex.org/W2004336806","https://openalex.org/W2006395211","https://openalex.org/W2011387411","https://openalex.org/W2014465300","https://openalex.org/W2035774625","https://openalex.org/W2041059091","https://openalex.org/W2042426695","https://openalex.org/W2048729985","https://openalex.org/W2053966966","https://openalex.org/W2082804915","https://openalex.org/W2094650250","https://openalex.org/W2104197095","https://openalex.org/W2148642357","https://openalex.org/W2155547400","https://openalex.org/W2156393948","https://openalex.org/W2160009442","https://openalex.org/W2332151900","https://openalex.org/W2608666998","https://openalex.org/W3107952763","https://openalex.org/W3137256632","https://openalex.org/W4241955075","https://openalex.org/W4254977685","https://openalex.org/W4300926135"],"related_works":["https://openalex.org/W3020555194","https://openalex.org/W2793256277","https://openalex.org/W4360958759","https://openalex.org/W2348235448","https://openalex.org/W2116218375","https://openalex.org/W2389906634","https://openalex.org/W176898926","https://openalex.org/W288156810","https://openalex.org/W1562406979","https://openalex.org/W2962797788"],"abstract_inverted_index":{"Abstract":[0],"We":[1,61,334],"study":[2],"the":[3,8,11,41,64,68,72,135,139,146,151,182,190,210,250,313,336],"behavior":[4],"at":[5],"infinity":[6],"of":[7,15,30,43,67,77,138,142,150,181,192,214,249,254,344],"farthest":[9,152,211,251],"and":[10,81,172,257,342],"higher-order":[12],"Voronoi":[13,75,153,183,212,252],"diagram":[14,76,154,184,213,253],"n":[16,78],"line":[17,79,255],"segments":[18,80,256],"or":[19],"lines":[20,82,226,258],"in":[21,227,262],"a":[22,37,187,207],"d":[23,147,179],"-dimensional":[24,148],"Euclidean":[25],"space.":[26],"The":[27,246],"unbounded":[28,140,194],"parts":[29],"these":[31,143],"diagrams":[32],"can":[33,259],"be":[34,260],"encoded":[35],"by":[36],"Gaussian":[38,69,201,247],"map":[39,70,248],"on":[40,199],"sphere":[42],"directions":[44],"$$\\mathbb":[45],"{S}^{d-1}$$":[46],"<mml:math":[47,86,118,159,218,233,266,292,304,321],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">":[48,87,119,160,219,234,267,293,305,322],"<mml:msup>":[49,101,271,326],"<mml:mrow>":[50,53,88,92,103,120,161,220,235,268,273,280,294,306,323],"<mml:mi>S</mml:mi>":[51],"</mml:mrow>":[52,57,100,107,110,129,167,224,242,277,284,286,298,310,331],"<mml:mi>d</mml:mi>":[54,104,163,274,295,307],"<mml:mo>-</mml:mo>":[55,97,105,122,164,239,275],"<mml:mn>1</mml:mn>":[56,106,127,165,240,276],"</mml:msup>":[58,108,278,329],"</mml:math>":[59,111,130,168,225,243,287,299,311,332],".":[60,131,333],"show":[62],"that":[63],"combinatorial":[65,136],"complexity":[66,137],"for":[71,116,289],"order-":[73],"k":[74],"is":[83,114,170,185,203],"$$O(\\min":[84],"\\{k,n-k\\}n^{d-1})$$":[85],"<mml:mi>O</mml:mi>":[89,125,269],"<mml:mo>(</mml:mo>":[90,126,162,237,270,281,325],"<mml:mo>min</mml:mo>":[91],"<mml:mo>{</mml:mo>":[93],"<mml:mi>k</mml:mi>":[94,98,123],"<mml:mo>,</mml:mo>":[95],"<mml:mi>n</mml:mi>":[96,102,121,221,236,238,272,282,327],"<mml:mo>}</mml:mo>":[99],"<mml:mo>)</mml:mo>":[109,128,166,241,283,285,330],",":[112,300,312],"which":[113],"tight":[115],"$$n-k=O(1)$$":[117],"<mml:mo>=</mml:mo>":[124,308],"This":[132],"exactly":[133,231],"reflects":[134],"features":[141],"diagrams.":[144],"All":[145],"cells":[149],"are":[155],"unbounded,":[156],"its":[157,193,200],"$$(d-1)$$":[158],"-skeleton":[169],"connected,":[171],"it":[173],"does":[174],"not":[175,204],"have":[176],"tunnels.":[177],"A":[178],"-cell":[180],"called":[186],"tunnel":[188],"if":[189,302],"set":[191],"directions,":[195],"represented":[196],"as":[197,346],"points":[198,345],"map,":[202],"connected.":[205],"In":[206],"three-dimensional":[208,244],"space,":[209],"$$n":[215],"\\ge":[216],"2$$":[217],"<mml:mo>\u2265</mml:mo>":[222,296],"<mml:mn>2</mml:mn>":[223,328],"general":[228],"position":[229],"has":[230],"$$n(n-1)$$":[232],"cells.":[245],"constructed":[261],"$$O(n^{d-1}":[263],"\\alpha":[264],"(n))$$":[265],"<mml:mi>\u03b1</mml:mi>":[279],"time,":[288],"$$d\\ge":[290],"4$$":[291],"<mml:mn>4</mml:mn>":[297],"while":[301],"$$d=3$$":[303],"<mml:mn>3</mml:mn>":[309],"time":[314],"drops":[315],"to":[316,339],"worst-case":[317],"optimal":[318],"$$\\Theta":[319],"(n^2)$$":[320],"<mml:mi>\u0398</mml:mi>":[324],"extend":[335],"obtained":[337],"results":[338],"bounded":[340],"polyhedra":[341],"clusters":[343],"sites.":[347]},"counts_by_year":[],"updated_date":"2026-06-13T06:13:01.061226","created_date":"2025-10-10T00:00:00"}
