{"id":"https://openalex.org/W2054037105","doi":"https://doi.org/10.1017/s0963548312000478","title":"Improved Mixing Time Bounds for the Thorp Shuffle","display_name":"Improved Mixing Time Bounds for the Thorp Shuffle","publication_year":2012,"publication_date":"2012-12-07","ids":{"openalex":"https://openalex.org/W2054037105","doi":"https://doi.org/10.1017/s0963548312000478","mag":"2054037105"},"language":"en","primary_location":{"id":"doi:10.1017/s0963548312000478","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0963548312000478","pdf_url":null,"source":{"id":"https://openalex.org/S7291079","display_name":"Combinatorics Probability Computing","issn_l":"0963-5483","issn":["0963-5483","1469-2163"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorics, Probability and 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/A5044639892","display_name":"Ben Morris","orcid":"https://orcid.org/0000-0002-9850-6071"},"institutions":[{"id":"https://openalex.org/I84218800","display_name":"University of California, Davis","ror":"https://ror.org/05rrcem69","country_code":"US","type":"education","lineage":["https://openalex.org/I84218800"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"BEN MORRIS","raw_affiliation_strings":["Department of Mathematics, University of California, Davis, CA 95616, USA (e-mail:","Department of mathematics, university of california, davis, ca 95616, usa e-mail: morris@math.ucdavis.edu#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of California, Davis, CA 95616, USA (e-mail:","institution_ids":["https://openalex.org/I84218800"]},{"raw_affiliation_string":"Department of mathematics, university of california, davis, ca 95616, usa e-mail: morris@math.ucdavis.edu#TAB#","institution_ids":["https://openalex.org/I84218800"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5044639892"],"corresponding_institution_ids":["https://openalex.org/I84218800"],"apc_list":{"value":3550,"currency":"USD","value_usd":3550},"apc_paid":null,"fwci":0.9685,"has_fulltext":false,"cited_by_count":15,"citation_normalized_percentile":{"value":0.7978016,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"22","issue":"1","first_page":"118","last_page":"132"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11797","display_name":"graph theory and CDMA systems","score":0.9998000264167786,"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"}},"topics":[{"id":"https://openalex.org/T11797","display_name":"graph theory and CDMA systems","score":0.9998000264167786,"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/T10951","display_name":"Cryptographic Implementations and Security","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/T11130","display_name":"Coding theory and cryptography","score":0.9993000030517578,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/shuffling","display_name":"Shuffling","score":0.9527641534805298},{"id":"https://openalex.org/keywords/mixing","display_name":"Mixing (physics)","score":0.6935848593711853},{"id":"https://openalex.org/keywords/pile","display_name":"Pile","score":0.6000668406486511},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.5470455884933472},{"id":"https://openalex.org/keywords/deck","display_name":"Deck","score":0.44809865951538086},{"id":"https://openalex.org/keywords/binary-number","display_name":"Binary number","score":0.410087525844574},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.39852944016456604},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.37758708000183105},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.37756288051605225},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.2823165953159332},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.2592591643333435},{"id":"https://openalex.org/keywords/engineering","display_name":"Engineering","score":0.24495482444763184},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.1882232427597046},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.136004239320755},{"id":"https://openalex.org/keywords/structural-engineering","display_name":"Structural engineering","score":0.10621017217636108},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.10143938660621643}],"concepts":[{"id":"https://openalex.org/C167927819","wikidata":"https://www.wikidata.org/wiki/Q1930567","display_name":"Shuffling","level":2,"score":0.9527641534805298},{"id":"https://openalex.org/C138777275","wikidata":"https://www.wikidata.org/wiki/Q6884054","display_name":"Mixing (physics)","level":2,"score":0.6935848593711853},{"id":"https://openalex.org/C119560385","wikidata":"https://www.wikidata.org/wiki/Q7193850","display_name":"Pile","level":2,"score":0.6000668406486511},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5470455884933472},{"id":"https://openalex.org/C2778966251","wikidata":"https://www.wikidata.org/wiki/Q208123","display_name":"Deck","level":2,"score":0.44809865951538086},{"id":"https://openalex.org/C48372109","wikidata":"https://www.wikidata.org/wiki/Q3913","display_name":"Binary number","level":2,"score":0.410087525844574},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.39852944016456604},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.37758708000183105},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.37756288051605225},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.2823165953159332},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.2592591643333435},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.24495482444763184},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.1882232427597046},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.136004239320755},{"id":"https://openalex.org/C66938386","wikidata":"https://www.wikidata.org/wiki/Q633538","display_name":"Structural engineering","level":1,"score":0.10621017217636108},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.10143938660621643},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1017/s0963548312000478","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0963548312000478","pdf_url":null,"source":{"id":"https://openalex.org/S7291079","display_name":"Combinatorics Probability Computing","issn_l":"0963-5483","issn":["0963-5483","1469-2163"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorics, Probability and Computing","raw_type":"journal-article"},{"id":"pmh:oai:escholarship.org/ark:/13030/qt9nw23866","is_oa":false,"landing_page_url":"https://escholarship.org/uc/item/9nw23866","pdf_url":null,"source":{"id":"https://openalex.org/S4306400115","display_name":"eScholarship (California Digital Library)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I2801248553","host_organization_name":"California Digital Library","host_organization_lineage":["https://openalex.org/I2801248553"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"article"},{"id":"pmh:qt9nw23866","is_oa":false,"landing_page_url":"http://www.escholarship.org/uc/item/9nw23866","pdf_url":null,"source":{"id":"https://openalex.org/S4306400115","display_name":"eScholarship (California Digital Library)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I2801248553","host_organization_name":"California Digital Library","host_organization_lineage":["https://openalex.org/I2801248553"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Morris, Ben. (2009). Improved mixing time bounds for the Thorp shuffle. UC Davis: Department of Mathematics. Retrieved from: http://www.escholarship.org/uc/item/9nw23866","raw_type":"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/W5776283","https://openalex.org/W124928283","https://openalex.org/W1491523234","https://openalex.org/W1551139065","https://openalex.org/W1552402248","https://openalex.org/W1559083062","https://openalex.org/W1559931973","https://openalex.org/W1593510332","https://openalex.org/W1972464518","https://openalex.org/W1976418263","https://openalex.org/W2029105598","https://openalex.org/W2046473466","https://openalex.org/W2053454826","https://openalex.org/W2056389572","https://openalex.org/W2069480672","https://openalex.org/W2077300005","https://openalex.org/W2099111195","https://openalex.org/W2120997209","https://openalex.org/W2123198691","https://openalex.org/W2475786860","https://openalex.org/W2568739026","https://openalex.org/W4205463258","https://openalex.org/W6667883438"],"related_works":["https://openalex.org/W4213239787","https://openalex.org/W4377131110","https://openalex.org/W2045745654","https://openalex.org/W2114337652","https://openalex.org/W4388857216","https://openalex.org/W2046590706","https://openalex.org/W4390419005","https://openalex.org/W2503156831","https://openalex.org/W2060171988","https://openalex.org/W3039689411"],"abstract_inverted_index":{"E.":[0],"Thorp":[1,83,118],"introduced":[2],"the":[3,9,16,28,32,37,42,52,78,82,90,105,117],"following":[4],"card":[5,30],"shuffling":[6],"model.":[7],"Suppose":[8],"number":[10],"of":[11,44,75,81,94,113],"cards":[12],"is":[13],"even.":[14],"Cut":[15],"deck":[17],"into":[18],"two":[19],"equal":[20],"piles,":[21],"then":[22],"interleave":[23],"them":[24],"as":[25],"follows.":[26],"Choose":[27],"first":[29],"from":[31,36,51],"left":[33],"pile":[34,39],"or":[35],"right":[38],"according":[40],"to":[41,108],"outcome":[43],"a":[45,97,110],"fair":[46],"coin":[47,61],"flip.":[48],"Then":[49],"choose":[50],"other":[53],"pile.":[54],"Continue":[55],"this":[56],"way,":[57],"flipping":[58],"an":[59,72,101],"independent":[60],"for":[62,77],"each":[63],"pair,":[64],"until":[65],"both":[66],"piles":[67],"are":[68],"empty.":[69],"We":[70],"prove":[71],"upper":[73],"bound":[74,93,103],"O(d3)":[76],"mixing":[79],"time":[80,106],"shuffle":[84],"with":[85],"2d":[86],"cards,":[87],"improving":[88],"on":[89,104],"best":[91],"known":[92],"O(d4).":[95],"As":[96],"consequence,":[98],"we":[99],"obtain":[100],"improved":[102],"required":[107],"encrypt":[109],"binary":[111],"message":[112],"length":[114],"d":[115],"using":[116],"shuffle.":[119]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":2},{"year":2017,"cited_by_count":2},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":1}],"updated_date":"2026-08-26T07:47:46.906454","created_date":"2025-10-10T00:00:00"}
