[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83771-en":3,"doc-seo-83771-105":30,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83771,4398048950312,"Violet","https://ap-avatar.wpscdn.com/avatar/400002538284de19e3c?_k=1778320343897328908",8,"Research & Report","Quasipolynomial Trace Reconstruction","Quasipolynomial trace reconstruction enables recovery of an unknown n-bit string x from a quasipolynomial number of deletion traces under the deletion channel with retention probability p. The result holds for any p at least inverse polylogarithmic in n, giving a bound of e^{p^{-7/3}(log_2 n)^c} traces for n>1. The analysis bypasses local statistical query barriers by exploiting global structure, reducing the problem to distinguishing traces from x versus y via iteratively “zooming out” around the first discrepancy.","arXiv :2607 .04073v 1 [ cs .DS] 5 Jul 2026  \nQuasipolynomial Trace Reconstruction  \nArnav Burudgunte, Paul Valiant, Hongao Wang  \nPurdue University  \nJuly 7, 2026  \nAbstract  \nWe show that trace reconstruction on n-bit strings is possible using a quasipolynomial number of traces, for any retention probability p that is at least inverse polylogarithmic in n.  \n1 Introduction  \nGiven an n-bit string x, a deletion channel with retention probability p deletes each bit of x with probability 1 − p and returns the remaining bits, which is called a trace. The trace reconstruction problem asks, how many traces from an unknown string x are needed to reconstruct x?  \nThis problem relates to some of the most fundamental issues in information theory, and has been studied—along with many proposed variants—for decades, both for its own sake and for the sake of applications. See Section 1 .2 for a discussion of related work.  \nFor constant retention probability p, the best lower bound says that ˜Ω(n3/2) traces are necessary˜ [Cha21a]; the best previously known upper bound is exponentially higher, saying that  \nexp(O(n1/5)) traces suffice [Cha21b] .  \nWe substantially improve our algorithmic understanding of trace reconstruction, showing aquasipolynomial upper bound.  \nTheorem. There exists a constant c > 0 such that for any retention probability p > 0, trace reconstruction on n > 1 bit strings can be done from ep−7/3(log2 n)c traces.  \nOur techniques bypass known barriers for the trace reconstruction problem. The strongest lower bound for the standard setting of trace reconstruction is the “local statistical query (SQ)”bound of [CDLS24], which we briefly introduce. An ℓ-local query specifies a function f on ℓ consecutive bits of a trace U, to which an oracle responds with a δ-accurate estimate of the expected value of f across traces from the unknown string. ˜ The surprising lower bound from this work is: any˜ trace reconstruction algorith˜m that makes O (n1/5)-local queries must have tolerance δ = ex˜ p(−Ω(n1/5)) . Further, the exp(O(n1/5)) upper bound of [Cha21b] ˜can be reinterpreted in a  \nO (n1/5)-local guise [CDLS24] . Thus, any algorithm that beats the exp(O(n1/5)) upper bound must take advantage of global structure in traces, in a way that previous algorithms have not.  \nOne of the simplest global queries asks: what is the expected value of the product of bits s 1 , ... , sk of a trace U ∼ Delp(x) . We call this a kth order statistic, and will denote it xp(s) (see Definition 3) . Our analysis will show how to identify a string from its polylog(n) order statistics.  \nIt is well known that, up to O (n) factors in sample complexity, the problem of recovering an unknown n-bit string x is equivalent to the problem of distinguishing a given pair x, y ∈ {0, 1}n , given repeated traces from one of the strings. It is often simpler to think of trace reconstruction as synonymous with this problem of distinguishing x from y.  \nOverall, our approach to distinguishing traces from x versus y can be described as “zooming out around the point d of first discrepancy between x, y.” Trace reconstruction from very short strings is easy, even with exponential approaches, so if we artificially pretend we had traces from a tiny window around d, then it is easy to find statistics s that strongly distinguish x from y, even simple statistics with k = 1 that look at the mean value of a single bit of the trace. Our idea is to iteratively transform a statistic s that only works on traces from a very local window around d into a new statistic S that survives being used on a more zoomed-out window around d. We thus iteratively transform a trivial k = 1 local statistic into a successively more global statistic of higher order until we have ultimately recovered a statistic that distinguishes x from y on traces from the entire string.  \nIntuitively, given a window of size R, the trace will produce Bin (R, p) bits, and thus, roughly, the location of individual bits will","cbCaiqUHPOtH85XU","https://ap.wps.com/l/cbCaiqUHPOtH85XU","pdf",630066,5,1,46,"English","en",105,"# Introduction\n## Trace reconstruction problem and bounds\n## Local statistical query barrier and global structure\n# Key idea: zooming out around the first discrepancy\n## Order statistics and distinguishing power\n## Blurring, induction, and multiple reference alignment analogy\n## Three-point test and reexpressing triple products","[{\"question\":\"What is the trace reconstruction problem addressed in this work?\",\"answer\":\"Given an unknown n-bit string x and traces produced by a deletion channel with retention probability p, the task is to determine how many traces are required to reconstruct x.\"},{\"question\":\"What is the main quantitative result for the number of traces?\",\"answer\":\"The paper proves that trace reconstruction on n\\u003e1 bit strings can be performed using e^{p^{-7/3}(log_2 n)^c} traces for any retention probability p\\u003e0 (with the stated regime tied to inverse polylogarithmic dependence).\"},{\"question\":\"Why are local statistical query lower bounds considered a barrier, and how does this work overcome it?\",\"answer\":\"Prior lower bounds based on local SQ queries imply that any algorithm relying only on O(n^{1/5})-local queries requires extremely small tolerance. This work improves the upper bound by leveraging global structure rather than only local statistics.\"}]",1784190318,116,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":28},"quasipolynomial-trace-reconstruction","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/quasipolynomial-trace-reconstruction/83771/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-27","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What is the trace reconstruction problem addressed in this work?","Question",{"text":76,"@type":77},"Given an unknown n-bit string x and traces produced by a deletion channel with retention probability p, the task is to determine how many traces are required to reconstruct x.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"What is the main quantitative result for the number of traces?",{"text":81,"@type":77},"The paper proves that trace reconstruction on n>1 bit strings can be performed using e^{p^{-7/3}(log_2 n)^c} traces for any retention probability p>0 (with the stated regime tied to inverse polylogarithmic dependence).",{"name":83,"@type":74,"acceptedAnswer":84},"Why are local statistical query lower bounds considered a barrier, and how does this work overcome it?",{"text":85,"@type":77},"Prior lower bounds based on local SQ queries imply that any algorithm relying only on O(n^{1/5})-local queries requires extremely small tolerance. This work improves the upper bound by leveraging global structure rather than only local statistics.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":20,"slug":138},19,"General","general"]