[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83396-en":3,"doc-seo-83396-105":29,"detail-sidebar-cat-0-en-105":91},{"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":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},83396,13056703020460,"Valentina","https://ap-avatar.wpscdn.com/avatar/be000253dac470eee5d?_k=1778207105932848923",8,"Research & Report","Algorithms and Indexing Lower Bounds for Variable String Matching","Generalized degenerate (GD) string matching considers a solid pattern P and a text T formed by concatenating one string from each segment of T, where all strings inside a segment share a common length (the segment width). The work gives a O(N√m)-time algorithm for GD matching, placing the problem on the subquadratic side of a fine-grained complexity boundary. It also develops indexed results: an O(nm) query approach for elastic-degenerate (ED) style indices adapted to GD, and strong SETH-based and k-clique–based conditional lower bounds limiting faster query times.","arXiv :2607 .08566v 1 [ cs .DS] 9 Jul 2026  \nAlgorithms and Indexing Lower Bounds for Variable String  \nMatching  \nEstéban Gabory∗  \nInstitute of Computer Science, University of Wrocław, Wrocław, Poland  \nAbstract  \nA generalized degenerate string (GD ) is a sequence T = T1 ···Tn of nonempty finite sets of strings (called segments) such that all strings within each segment have the same length, called the width of the segment. We study pattern matching in generalized degenerate strings: given a solid string P and a GD string T , decide whether P occurs in T , that is, whether P is a substring of a string obtained by concatenating one string from each segment of T. This problem was identified by Ascone et al. (WABI 2024) as the main remaining open case in the fine-grained complexity of pattern matching on variable strings, a family of string representations for sets of similar strings with different levels of generality. It is the remaining boundary case between the variants admitting near-linear-time algorithmsand˜ those with quadratic conditional lower bounds under SETH. In this paper, we show  \na O (N√m)-time algorithm for GD string matching, where N is the total size of T and m = |P|, placing the problem on the subquadratic side of this boundary.  \nWe complement this with upper and lower bounds for the indexed setting. For elasticdegenerate strings (ED ), where the equal-width condition is dropped, Gibney (SPIRE 2020) showed that one can answer pattern matching queries in O (nm2 ) time where m is the length of the pattern and n is the number of segments of the ED string, after linear-time preprocessing. We show that this index can be adapted to GD strings to answer queries in O (nm) time. We observe that an existing SETH-based reduction from the same paper already applies to GD strings and rules out indices with polynomial-time preprocessing and with query time O (n1−εmO(1) + m) for any ε > 0. We then prove that, under the k-clique conjecture, no combinatorial index with polynomial preprocessing time can answer queries in O (nO(1)m 1−ε + m) time for any ε > 0. For ED strings, we prove under the same conjecture that no combinatorial index with polynomial preprocessing can answer queries in O (nO(1)m2−ε ) time for any ε > 0, matching the quadratic dependence on m in Gibney’supper bound. Finally, assuming the OMv conjecture, we show that there is no data structure that, after polynomial-time preprocessing of a set of strings and a pattern, answers activeprefix queries on a bit vector of length m in O (m2−ε ) time for any ε > 0; such queries are the standard bottleneck in elastic-degenerate pattern matching. Taken together, these lower bounds suggest that improving the query time for indexed ED pattern matching below O (nO(1)m2 ) would require both non-combinatorial techniques and an approach that avoids using active-prefix queries as the central bottleneck.  \n1 Introduction  \nMotivation and Background. String matching (or pattern matching) is a fundamental task in computer science. Given a text and a pattern, one asks whether the pattern occurs in the text, or asks to count or report all occurrences. This simplest setting can be solved in linear time [9] . This line of research has since been extended to more complex settings, such as approximate matching [21, 22, 8, 14, 7], or pattern matching on different data structures [19, 23, 18, 16, 2, 5] .  \n∗ Supported by the Polish National Science Centre grant number 2023/51/B/ST6/01505 .  \n􀀚  \n􀀚 AC 􀀛· 􀀈 G 􀀉· 􀀚  \nAC  \nGT  \n􀀛· 􀀚  \n(a)  \nA  \nC  \n􀀛· 􀀚  \n(c)  \nTA 􀀛  \nTGA  \nCAG  \n􀀛  \n􀀚 ACGT 􀀛· 􀀚 TG  \nCA  \n(b)  \n􀀚 ACG 􀀛· 􀀚 TεGA (d)  \n􀀛· 􀀚  \n􀀛· 􀀚  \nAA  \nGC  \n􀀛  \nC  \nAGT  \n􀀛  \nFigure 1: Four types of variable strings: (a) degenerate strings have width 1 in every segment;(b) k-degenerate strings have one fixed width k; (c) GD strings allow the width to vary across segments; (d) ED strings also allow different string lengths inside a segment.  \nA natural generalization is to replace each posi","cbCaiqfAMEQZX3Dm","https://ap.wps.com/l/cbCaiqfAMEQZX3Dm","pdf",677573,1,13,"English","en",105,"# Abstract\n# Introduction\n## Motivation and Background","[{\"question\":\"What is a generalized degenerate (GD) string in this paper?\",\"answer\":\"A GD string is a sequence of nonempty segments, where every string inside a segment has the same length (the segment width). Pattern matching checks whether a solid pattern P occurs as a substring of a concatenation choosing one string from each segment.\"},{\"question\":\"What algorithm is proposed for GD string matching?\",\"answer\":\"The paper presents an O(N√m)-time algorithm for GD string matching, where N is the total size of T and m=|P|.\"},{\"question\":\"What do the paper’s conditional lower bounds say about faster indexed queries?\",\"answer\":\"Using SETH-based reductions and the k-clique conjecture, the paper rules out combinatorial indices with polynomial preprocessing that achieve query time improvements beyond specified thresholds, and it further shows OMv-based limits for active-prefix queries in the elastic-degenerate setting.\"}]",1784187227,33,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"algorithms-and-indexing-lower-bounds-for-variable-string-matching","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/algorithms-and-indexing-lower-bounds-for-variable-string-matching/83396/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is a generalized degenerate (GD) string in this paper?","Question",{"text":75,"@type":76},"A GD string is a sequence of nonempty segments, where every string inside a segment has the same length (the segment width). Pattern matching checks whether a solid pattern P occurs as a substring of a concatenation choosing one string from each segment.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What algorithm is proposed for GD string matching?",{"text":80,"@type":76},"The paper presents an O(N√m)-time algorithm for GD string matching, where N is the total size of T and m=|P|.",{"name":82,"@type":73,"acceptedAnswer":83},"What do the paper’s conditional lower bounds say about faster indexed queries?",{"text":84,"@type":76},"Using SETH-based reductions and the k-clique conjecture, the paper rules out combinatorial indices with polynomial preprocessing that achieve query time improvements beyond specified thresholds, and it further shows OMv-based limits for active-prefix queries in the elastic-degenerate setting.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]