{"id":"https://openalex.org/W2059541050","doi":"https://doi.org/10.1137/060669474","title":"On Approximating the Depth and Related Problems","display_name":"On Approximating the Depth and Related Problems","publication_year":2008,"publication_date":"2008-01-01","ids":{"openalex":"https://openalex.org/W2059541050","doi":"https://doi.org/10.1137/060669474","mag":"2059541050"},"language":"en","primary_location":{"id":"doi:10.1137/060669474","is_oa":false,"landing_page_url":"https://doi.org/10.1137/060669474","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/A5042232059","display_name":"Boris Aronov","orcid":"https://orcid.org/0000-0003-3110-4702"},"institutions":[{"id":"https://openalex.org/I57206974","display_name":"New York University","ror":"https://ror.org/0190ak572","country_code":"US","type":"education","lineage":["https://openalex.org/I57206974"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Boris Aronov","raw_affiliation_strings":["New York University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"New York University","institution_ids":["https://openalex.org/I57206974"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5040577923","display_name":"Sariel Har-Peled","orcid":"https://orcid.org/0000-0003-2638-9635"},"institutions":[{"id":"https://openalex.org/I157725225","display_name":"University of Illinois Urbana-Champaign","ror":"https://ror.org/047426m28","country_code":"US","type":"education","lineage":["https://openalex.org/I157725225"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Sariel Har-Peled","raw_affiliation_strings":["University of Illinois at Urbana Champaign"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Illinois at Urbana Champaign","institution_ids":["https://openalex.org/I157725225"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":33.363,"has_fulltext":false,"cited_by_count":123,"citation_normalized_percentile":{"value":0.99733028,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"38","issue":"3","first_page":"899","last_page":"921"},"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/T11106","display_name":"Data Management and Algorithms","score":0.9997000098228455,"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"}},{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9994000196456909,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/range","display_name":"Range (aeronautics)","score":0.663652777671814},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6315567493438721},{"id":"https://openalex.org/keywords/point","display_name":"Point (geometry)","score":0.5973230004310608},{"id":"https://openalex.org/keywords/linear-programming","display_name":"Linear programming","score":0.47902384400367737},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.46970126032829285},{"id":"https://openalex.org/keywords/sampling","display_name":"Sampling (signal processing)","score":0.4679234027862549},{"id":"https://openalex.org/keywords/plane","display_name":"Plane (geometry)","score":0.46416524052619934},{"id":"https://openalex.org/keywords/time-complexity","display_name":"Time complexity","score":0.45345398783683777},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.45125919580459595},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.44773030281066895},{"id":"https://openalex.org/keywords/counting-problem","display_name":"Counting problem","score":0.43211841583251953},{"id":"https://openalex.org/keywords/approximation-algorithm","display_name":"Approximation algorithm","score":0.41606661677360535},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.321106493473053},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.2966630458831787},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.11267834901809692}],"concepts":[{"id":"https://openalex.org/C204323151","wikidata":"https://www.wikidata.org/wiki/Q905424","display_name":"Range (aeronautics)","level":2,"score":0.663652777671814},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6315567493438721},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.5973230004310608},{"id":"https://openalex.org/C41045048","wikidata":"https://www.wikidata.org/wiki/Q202843","display_name":"Linear programming","level":2,"score":0.47902384400367737},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.46970126032829285},{"id":"https://openalex.org/C140779682","wikidata":"https://www.wikidata.org/wiki/Q210868","display_name":"Sampling (signal processing)","level":3,"score":0.4679234027862549},{"id":"https://openalex.org/C17825722","wikidata":"https://www.wikidata.org/wiki/Q17285","display_name":"Plane (geometry)","level":2,"score":0.46416524052619934},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.45345398783683777},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.45125919580459595},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.44773030281066895},{"id":"https://openalex.org/C16592021","wikidata":"https://www.wikidata.org/wiki/Q5177154","display_name":"Counting problem","level":2,"score":0.43211841583251953},{"id":"https://openalex.org/C148764684","wikidata":"https://www.wikidata.org/wiki/Q621751","display_name":"Approximation algorithm","level":2,"score":0.41606661677360535},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.321106493473053},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2966630458831787},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.11267834901809692},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C31972630","wikidata":"https://www.wikidata.org/wiki/Q844240","display_name":"Computer vision","level":1,"score":0.0},{"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/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C106131492","wikidata":"https://www.wikidata.org/wiki/Q3072260","display_name":"Filter (signal processing)","level":2,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/060669474","is_oa":false,"landing_page_url":"https://doi.org/10.1137/060669474","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":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":23,"referenced_works":["https://openalex.org/W1965996575","https://openalex.org/W1980092773","https://openalex.org/W1995232397","https://openalex.org/W2002581612","https://openalex.org/W2005538657","https://openalex.org/W2010790640","https://openalex.org/W2015858208","https://openalex.org/W2027277859","https://openalex.org/W2027387410","https://openalex.org/W2041935575","https://openalex.org/W2049083558","https://openalex.org/W2094650250","https://openalex.org/W2103893986","https://openalex.org/W2139225572","https://openalex.org/W2153204081","https://openalex.org/W2155229861","https://openalex.org/W2160775966","https://openalex.org/W2220451813","https://openalex.org/W3144422206","https://openalex.org/W4213071546","https://openalex.org/W4238284510","https://openalex.org/W4242657188","https://openalex.org/W4254977685"],"related_works":["https://openalex.org/W2962917430","https://openalex.org/W2107423006","https://openalex.org/W2028468530","https://openalex.org/W3149705110","https://openalex.org/W4299801389","https://openalex.org/W2574604093","https://openalex.org/W2775068415","https://openalex.org/W4287424652","https://openalex.org/W1903284337","https://openalex.org/W2767824041"],"abstract_inverted_index":{"We":[0,54],"study":[1],"the":[2,33,61,64,73,81,94],"question":[3],"of":[4,12,67],"finding":[5],"a":[6,16,43],"deepest":[7,34],"point":[8,35],"in":[9,80],"an":[10],"arrangement":[11],"regions":[13],"and":[14,97],"provide":[15],"fast":[17,44],"algorithm":[18,45],"for":[19,46],"this":[20,30,57],"problem":[21,31],"using":[22],"random":[23],"sampling,":[24],"showing":[25],"it":[26],"sufficient":[27],"to":[28,59],"solve":[29],"when":[32],"is":[36],"shallow.":[37],"This":[38],"implies,":[39],"among":[40],"other":[41],"results,":[42],"approximately":[47],"solving":[48],"linear":[49],"programming":[50],"problems":[51],"with":[52],"violations.":[53],"also":[55],"use":[56],"technique":[58],"approximate":[60,88],"disk":[62],"covering":[63],"largest":[65],"number":[66],"red":[68],"points,":[69,75],"while":[70],"avoiding":[71],"all":[72],"blue":[74],"given":[76],"two":[77],"such":[78],"sets":[79],"plane.":[82],"Using":[83],"similar":[84],"techniques":[85],"implies":[86],"that":[87],"range":[89,102],"counting":[90],"queries":[91],"have":[92],"roughly":[93],"same":[95],"time":[96],"space":[98],"complexity":[99],"as":[100],"emptiness":[101],"queries.":[103]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":8},{"year":2019,"cited_by_count":3},{"year":2018,"cited_by_count":6},{"year":2017,"cited_by_count":7},{"year":2016,"cited_by_count":10},{"year":2015,"cited_by_count":9},{"year":2014,"cited_by_count":5},{"year":2013,"cited_by_count":8},{"year":2012,"cited_by_count":6}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}