[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84559-en":3,"doc-seo-84559-105":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},84559,8796095360427,"Lucas Martin","https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d",8,"Research & Report","Efficient LCE Queries and Lexicographic Minimizers on Sliding Suffix Trees","Efficient longest-common-extension (LCE) queries and lexicographic minimizer maintenance are developed for sliding-window suffix-tree indexing. The key challenge is that a sliding suffix tree is kept in an implicit, Ukkonen-style form where many suffixes of the current window end inside edges and lack explicit leaves. A periodic representative map folds any implicit suffix to an explicit suffix leaf in constant time, enabling constant-time LCE queries using dynamic LCA and leaf pointers, and supporting exact minimizers either via LCE-based comparison or via an LCE-free frontier-node construction with lexicographic order labels.","arXiv :2607 .00389v 1 [ cs .DS] 1 Jul 2026  \nEfficient LCE Queries and Lexicographic Minimizers on Sliding Suffix Trees  \nToshiharu Minematsu 1 and Shunsuke Inenaga2  \n1 Department of Information Science and Technology, Kyushu University  \n[minematsu.toshiharu.862@s.kyushu-u.ac.jp](minematsu.toshiharu.862@s.kyushu-u.ac.jp)  \n2 Department of Informatics, Kyushu University  \n[inenaga.shunsuke.380@m.kyushu-u.ac.jp](inenaga.shunsuke.380@m.kyushu-u.ac.jp)  \nAbstract. We study longest-common-extension (LCE) queries and lexicographic minimizer maintenance on the suffix tree of a sliding window.  \nThe main difficulty is that a sliding suffix tree is maintained in an implicit Ukkonen-style form: some suffixes of the current window are not represented by leaves. We show that the longest implicit suffix induces a periodic representative map that folds every implicit suffix to an explicit suffix leaf in constant time. Combined with leaf pointers and dynamic LCA, this yields a linear-space data structure with amortized constanttime window shifts and worst-case constant-time LCE queries over a constant-size alphabet. For minimizers, the LCE structure gives a direct exact solution, but it uses more machinery than fixed-depth comparisons require. We therefore give an LCE-free construction. It materializes each active depth-k locus as an auxiliary frontier node storing a frontier token.  \nSuffix-leaf handles are created immediately when leaves appear; handles of suffixes shorter than k stay inactive and are activated only when their k-mer becomes defined. Tokens are compared by dynamically maintained lexicographic leaf-order labels of representative handles, while k-mer positions whose suffixes are implicit are folded by the same representative map. The common monotone-deque layer over these token pointers then maintains exact lexicographic minimizers without hashing, LCE queries, or dynamic LCA.  \n1 Introduction  \nSuffix trees are classical indexes for exact string processing, and Ukkonen’s online construction is the standard starting point for maintaining them incrementally [17] . In a streaming setting, however, the object to be indexed is often only the mostrecent window, leading to the sliding suffix tree. This line of work goes back to the finite-window data-compression index of Fiala and Greene [6], was refined by Larsson [8], and was later revisited by Senft [15] . Leonard et al. simplified the maintenance of valid edge labels and leaf pointers [9] . Recent work on streaming sliding-window string indexing gives a complementary view of indexing the mostrecent window under streaming updates [2,3] . We use the sliding suffix tree of Brodnik and Jekovec as the underlying dynamic suffix tree, together with the leaf pointers and valid edge labels of Leonard et al. [4,9] . These primitives are  \n2 Minematsu and Inenaga  \nsufficient for the LCE data structure developed in Sections 3 and 4 . For the LCE-free minimizer data structure of Section 5, we additionally use the BP-linked suffix tree of Sumiyoshi et al. [16] together with a standard order-maintenance structure to compare representative suffix leaves in constant time.  \nThis paper studies how to use this representation for two comparison taskson the current window S: LCE queries and lexicographic minimizer maintenance. For positions i, j , LCE (i, j) is the longest common prefix length of S [i..] and S [j..] . For an integer k, the lexicographic minimizer is the leftmost occurrence of the smallest current k-mer. Minimizers were introduced as a sparse seed-selection mechanism for biological sequence comparison [13], and are closely related to winnowing schemes for local fingerprinting [14] . Their density and ordering have been studied extensively [12, 11] . Both tasks are immediate in a static suffix tree with a terminal symbol, but not in a sliding suffix tree without one: some suffixes are implicit, ending inside edges rather than at explicit suffix leaves. Thus LCE queries and k-mer c","cbCaikFXaJg3bfbo","https://ap.wps.com/l/cbCaikFXaJg3bfbo","pdf",585430,1,17,"English","en",105,"# Introduction\n## Sliding suffix trees and the implicit-suffix challenge\n## Representative map for implicit suffixes\n## LCE queries and minimizer maintenance approaches","[{\"question\":\"What problem does the paper address for sliding suffix trees?\",\"answer\":\"It studies LCE queries and lexicographic minimizer maintenance when indexing only the most recent sliding window. The main difficulty is that many suffixes in the implicit Ukkonen-style representation do not correspond to explicit suffix leaves.\"},{\"question\":\"How are implicit suffixes handled to enable constant-time operations?\",\"answer\":\"The paper introduces a periodic representative map that folds each implicit suffix to an explicit suffix leaf in constant time. This map then supports reductions to LCA queries on explicit leaves with truncation based on true suffix lengths.\"},{\"question\":\"How are lexicographic minimizers maintained without relying on LCE queries?\",\"answer\":\"An LCE-free method materializes active depth-k loci as auxiliary frontier nodes with frontier tokens. Tokens are compared using dynamically maintained lexicographic leaf-order labels of representative handles, with implicit suffixes folded by the same representative map, then using a monotone-deque layer to maintain exact minimizers.\"}]",1784196728,43,{"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},"efficient-lce-queries-and-lexicographic-minimizers-on-sliding-suffix-trees","",{"@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/efficient-lce-queries-and-lexicographic-minimizers-on-sliding-suffix-trees/84559/",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 problem does the paper address for sliding suffix trees?","Question",{"text":75,"@type":76},"It studies LCE queries and lexicographic minimizer maintenance when indexing only the most recent sliding window. The main difficulty is that many suffixes in the implicit Ukkonen-style representation do not correspond to explicit suffix leaves.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are implicit suffixes handled to enable constant-time operations?",{"text":80,"@type":76},"The paper introduces a periodic representative map that folds each implicit suffix to an explicit suffix leaf in constant time. This map then supports reductions to LCA queries on explicit leaves with truncation based on true suffix lengths.",{"name":82,"@type":73,"acceptedAnswer":83},"How are lexicographic minimizers maintained without relying on LCE queries?",{"text":84,"@type":76},"An LCE-free method materializes active depth-k loci as auxiliary frontier nodes with frontier tokens. Tokens are compared using dynamically maintained lexicographic leaf-order labels of representative handles, with implicit suffixes folded by the same representative map, then using a monotone-deque layer to maintain exact minimizers.","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"]