{"id":"https://openalex.org/W7204810967","doi":"https://doi.org/10.48550/arxiv.2608.27777","title":"Optimal exponential memory for sequential Euclidean connections: edge-power costs and phase transitions","display_name":"Optimal exponential memory for sequential Euclidean connections: edge-power costs and phase transitions","publication_year":2026,"publication_date":"2026-08-27","ids":{"openalex":"https://openalex.org/W7204810967","doi":"https://doi.org/10.48550/arxiv.2608.27777"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.27777","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.27777","pdf_url":"https://arxiv.org/pdf/2608.27777","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2608.27777","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5148422155","display_name":"Pedro M. M. de Castro","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"de Castro, Pedro M. M.","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5148422155"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11830","display_name":"Point processes and geometric inequalities","score":0.1906999945640564,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11830","display_name":"Point processes and geometric inequalities","score":0.1906999945640564,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.11169999837875366,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.09749999642372131,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/memory-parameter","display_name":"memory parameter","score":0.5600000023841858},{"id":"https://openalex.org/keywords/labeled-trees","display_name":"labeled trees","score":0.621999979019165},{"id":"https://openalex.org/keywords/phase-transitions","display_name":"phase transitions","score":0.6980000138282776},{"id":"https://openalex.org/keywords/local-maxima","display_name":"local maxima","score":0.6129999756813049},{"id":"https://openalex.org/keywords/worst-case-analysis","display_name":"worst-case analysis","score":0.43799999356269836}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7821999788284302},{"id":"https://openalex.org/C168773036","wikidata":"https://www.wikidata.org/wiki/Q264164","display_name":"Recursion (computer science)","level":2,"score":0.4765999913215637},{"id":"https://openalex.org/C189237950","wikidata":"https://www.wikidata.org/wiki/Q2500758","display_name":"Stationary point","level":2,"score":0.47130000591278076},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.4645000100135803},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.4553000032901764},{"id":"https://openalex.org/C15627037","wikidata":"https://www.wikidata.org/wiki/Q541961","display_name":"Classification of discontinuities","level":2,"score":0.4528999924659729},{"id":"https://openalex.org/C48103436","wikidata":"https://www.wikidata.org/wiki/Q599031","display_name":"State (computer science)","level":2,"score":0.4489000141620636},{"id":"https://openalex.org/C139676723","wikidata":"https://www.wikidata.org/wiki/Q1193832","display_name":"Sign (mathematics)","level":2,"score":0.4361000061035156},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4336000084877014},{"id":"https://openalex.org/C186633575","wikidata":"https://www.wikidata.org/wiki/Q845060","display_name":"Maxima and minima","level":2,"score":0.4309000074863434},{"id":"https://openalex.org/C13355873","wikidata":"https://www.wikidata.org/wiki/Q2920850","display_name":"Connection (principal bundle)","level":2,"score":0.42500001192092896},{"id":"https://openalex.org/C2681867","wikidata":"https://www.wikidata.org/wiki/Q690935","display_name":"Saddle point","level":2,"score":0.3831999897956848},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.3659000098705292},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.36559998989105225},{"id":"https://openalex.org/C11683690","wikidata":"https://www.wikidata.org/wiki/Q752726","display_name":"Asymptotic expansion","level":2,"score":0.3653999865055084},{"id":"https://openalex.org/C119047807","wikidata":"https://www.wikidata.org/wiki/Q752718","display_name":"Asymptotic analysis","level":2,"score":0.3474000096321106},{"id":"https://openalex.org/C91528185","wikidata":"https://www.wikidata.org/wiki/Q520952","display_name":"Maxima","level":3,"score":0.32710000872612},{"id":"https://openalex.org/C205147927","wikidata":"https://www.wikidata.org/wiki/Q752718","display_name":"Asymptotic analysis","level":2,"score":0.3248000144958496},{"id":"https://openalex.org/C159886148","wikidata":"https://www.wikidata.org/wiki/Q176645","display_name":"Markov process","level":2,"score":0.32280001044273376},{"id":"https://openalex.org/C2778755073","wikidata":"https://www.wikidata.org/wiki/Q10858537","display_name":"Scale (ratio)","level":2,"score":0.3174000084400177},{"id":"https://openalex.org/C98763669","wikidata":"https://www.wikidata.org/wiki/Q176645","display_name":"Markov chain","level":2,"score":0.3140999972820282},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.30880001187324524},{"id":"https://openalex.org/C91575142","wikidata":"https://www.wikidata.org/wiki/Q1971426","display_name":"Optimal control","level":2,"score":0.3086000084877014},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.3046000003814697},{"id":"https://openalex.org/C149288129","wikidata":"https://www.wikidata.org/wiki/Q185357","display_name":"Phase transition","level":2,"score":0.2994999885559082},{"id":"https://openalex.org/C95611797","wikidata":"https://www.wikidata.org/wiki/Q17502105","display_name":"Infimum and supremum","level":2,"score":0.29589998722076416},{"id":"https://openalex.org/C47458137","wikidata":"https://www.wikidata.org/wiki/Q911951","display_name":"Stationary state","level":2,"score":0.2948000133037567},{"id":"https://openalex.org/C60798267","wikidata":"https://www.wikidata.org/wiki/Q1226939","display_name":"Division (mathematics)","level":2,"score":0.29019999504089355},{"id":"https://openalex.org/C143724316","wikidata":"https://www.wikidata.org/wiki/Q312468","display_name":"Series (stratigraphy)","level":2,"score":0.2890999913215637},{"id":"https://openalex.org/C99888217","wikidata":"https://www.wikidata.org/wiki/Q1288707","display_name":"Stopping time","level":2,"score":0.2793000042438507},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.27730000019073486},{"id":"https://openalex.org/C120174047","wikidata":"https://www.wikidata.org/wiki/Q847073","display_name":"Euclidean distance","level":2,"score":0.2761000096797943},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.27140000462532043},{"id":"https://openalex.org/C44280652","wikidata":"https://www.wikidata.org/wiki/Q104837","display_name":"Phase (matter)","level":2,"score":0.25780001282691956}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.27777","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.27777","pdf_url":"https://arxiv.org/pdf/2608.27777","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2608.27777","doi":"https://doi.org/10.48550/arxiv.2608.27777","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.27777","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2608.27777","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.27777","pdf_url":"https://arxiv.org/pdf/2608.27777","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7204810967.pdf","grobid_xml":"https://content.openalex.org/works/W7204810967.grobid-xml"},"referenced_works_count":17,"referenced_works":["https://openalex.org/W2025845565","https://openalex.org/W2027943624","https://openalex.org/W2029437526","https://openalex.org/W2077586119","https://openalex.org/W2101027188","https://openalex.org/W2149560766","https://openalex.org/W2259315298","https://openalex.org/W2280241271","https://openalex.org/W2460736097","https://openalex.org/W2771742267","https://openalex.org/W2951738458","https://openalex.org/W2964333910","https://openalex.org/W3090797528","https://openalex.org/W4226249744","https://openalex.org/W4229980202","https://openalex.org/W4293282250","https://openalex.org/W7204435731"],"related_works":["https://openalex.org/W7204435731","https://openalex.org/W2037992074","https://openalex.org/W6940964482","https://openalex.org/W2101027188","https://openalex.org/W2029565994","https://openalex.org/W7099600471","https://openalex.org/W7052747512","https://openalex.org/W3002079151","https://openalex.org/W1968448539","https://openalex.org/W27726302"],"abstract_inverted_index":{"We":[0,65,117],"study":[1],"the":[2,6,11,32,41,67,70,73,88,91,105,119,123,136,140,150,166,177,181,194,201,233,259],"edge-power":[3,207],"cost":[4,208,235],"of":[5,69,72,189],"labelled":[7],"tree":[8],"generated":[9],"by":[10,36],"$\u03b3$-strategy,":[12],"a":[13,45,49,95,187,228,247,255],"constant-gain":[14],"rule":[15],"for":[16,107,115,213,242,250],"sequential":[17],"Euclidean":[18],"connections.":[19],"Starting":[20],"with":[21,158,227],"$x_0=p_0$,":[22],"each":[23],"input":[24,79,82,199],"point":[25,210],"$p_i$":[26],"is":[27,34,102,113,144,236],"attached":[28],"to":[29,238],"$x_{i-1}$,":[30],"and":[31,48,80,148,169,186,204,245,254],"state":[33],"updated":[35],"$x_i=\u03b3x_{i-1}+(1-\u03b3)p_i$.":[37],"Retaining":[38],"$x_i$":[39],"subdivides":[40],"insertion":[42],"segment":[43],"into":[44],"spine":[46],"edge":[47,74],"leaf":[50],"edge.":[51],"The":[52],"memory":[53],"parameter":[54,195,203],"$\u03b3$":[55],"controls":[56],"how":[57],"long":[58],"earlier":[59],"points":[60,86],"influence":[61],"subsequent":[62],"attachment":[63],"points.":[64],"minimize":[66],"sum":[68],"$\u03b1$-powers":[71],"lengths":[75],"under":[76],"independent":[77],"uniform":[78,85],"arbitrary":[81,198],"sequences.":[83],"For":[84,197],"in":[87,122],"unit":[89],"ball,":[90],"stationary":[92,156,182],"problem":[93],"has":[94],"transition":[96],"at":[97,104,139,180],"$\u03b1=1$.":[98],"Its":[99],"continuous":[100],"extension":[101],"minimized":[103],"boundary":[106],"$0<\u03b1\\leq1$,":[108],"while":[109],"every":[110,251],"global":[111],"minimizer":[112],"interior":[114],"$\u03b1>1$.":[116],"determine":[118],"finite":[120],"optimizer":[121],"joint":[124],"window":[125],"$\u03b1_N=1+\\varepsilon_N$,":[126],"$\\varepsilon_N\\log":[127],"N\\to\u03bb$.":[128],"Below":[129],"an":[130,154],"explicit":[131,155,212,223],"threshold":[132,141,151,164],"it":[133,152],"lies":[134],"on":[135],"$N^{-1/2}$":[137],"scale,":[138],"its":[142],"scale":[143],"$\\sqrt{\\log":[145],"N/(N\\log\\log":[146],"N)}$,":[147],"above":[149],"approaches":[153],"root":[157],"two":[159,244],"computable":[160],"corrections.":[161],"A":[162],"second":[163],"identifies":[165],"governing":[167],"correction,":[168],"differentiated":[170],"estimates":[171],"prove":[172],"eventual":[173],"uniqueness.":[174],"At":[175,215],"$\u03b1=3d+8$,":[176],"linear":[178],"coefficient":[179],"endpoint":[183],"changes":[184],"sign":[185],"branch":[188],"strict":[190],"local":[191],"maxima":[192],"enters":[193],"interval.":[196],"sequences,":[200],"optimal":[202],"asymptotic":[205,237],"worst-case":[206],"per":[209],"are":[211],"$0<\u03b1\\leq3$.":[214],"high":[216],"powers,":[217],"periodic":[218],"antipodal":[219],"block":[220],"inputs":[221],"give":[222],"lower":[224],"bounds":[225],"which,":[226],"separation":[229],"argument,":[230],"show":[231],"that":[232],"optimized":[234],"$2\\log2/\\log\u03b1$.":[239],"Exact":[240],"results":[241],"powers":[243],"four,":[246],"rational":[248],"recursion":[249],"even":[252],"power,":[253],"high-dimensional":[256],"expansion":[257],"complete":[258],"analysis.":[260]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-01T00:00:00"}