[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83558-en":3,"doc-seo-83558-105":30,"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":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},83558,34359740700684,"Finn","https://ap-avatar.wpscdn.com/avatar/1f400023980c374ae676?_k=1777273430885731487",8,"Research & Report","Tighter Bounds for Wheeler Determinization","Given a Wheeler NFA A, Wheeler determinization constructs a Wheeler DFA D that recognizes the same language. The paper measures graph size via nA, mA and nD, mD and builds on prior O(nA^3) algorithms. Using the Wheeler order of A, computed in O(mA log nA), it achieves an output-sensitive running time of O(nA + mA + nD + mD). The method speeds existing results by a factor nA^2/σ, yields a first linear-time algorithm for constant alphabet size, and proves tight bounds through worst-case input families.","arXiv :2607 .0 1007v 1 [ cs .DS] 1 Jul 2026  \nTighter Bounds for Wheeler Determinization  \nPhilip Bille 1[0000−0002−1120−5154], Inge Li Gørtz 1[0000−0002−8322−4952], Máximo Pérez-López 1[0009−0001−8507−4820], and Simon R. Tarnow 1[0009−0002−4293−6475]  \nTechnical University of Denmark, 2800 Lyngby, Denmark  \nAbstract. Given a Wheeler NFA A, the Wheeler determinization problem is to construct a Wheeler DFA D that accepts the same language as  \nA. We use the notation nA , mA for the number of vertices and edges of A, and equivalently nD , mD for D. Alanko et al. [3, 4] show that we can solve this problem in O (n3A) time. In this paper, we show how to improve the running time to O (nA + mA + nD + mD ) when given the Wheeler order of A (which can be computed in O (mA log nA ) with an algorithm by Becker et al. [6]) .  \nOur running time is a factor n2A/σ faster than the state of the art, where σ is the size of the alphabet. Furthermore, for σ = O(1) we have the first linear time algorithm for this problem. We show that our bound is tight for sorted inputs with any combination of n and σ, by giving a family of inputs for which our output D is minimum, and of maximum size Θ (nσ) .  \nKeywords: Wheeler graphs · Automata · Determinization.  \n1 Introduction  \nWheeler graphs [14] are a class of directed, labelled graphs that has received attention in both the compression community and bioinformatics. In compression, Wheeler graphs allow storing a (potentially infinite) set of strings, and support indexing for matching patterns P in O (|P|log σ) time, in contrast to the conditional lower bound of ω(mδ |P|β ) time (for δ \u003C 1 or β \u003C 1) required in general labelled graphs [12] . They have also been studied as an important subclass of finite automata [3, 4] . In the bioinformatics community, Wheeler graphs have been used for indexing genomic databases. The compressed index of the variation graph toolkit VG [15] is based on the Wheeler properties of the de Bruijn graph of the data, as well as other indices such as VARI [20] and Themisto [22] .  \nMost of the algorithmic work on Wheeler graphs has been described on deterministic Wheeler graphs, that is, graphs where no two edges with the same label leave the same vertex. Important examples are computing matching statistics [9], computing the LCP array [1], counting distinct k-mers [5], and minimizing the size [2] . Further work that extends results from Wheeler graphs to arbitrary finite automata also maintains the deterministic restriction [19, 10], or provides efficient algorithms only for deterministic automata [11] . It is therefore an important question whether these results can be extended to non-deterministic graphs, and under what conditions.  \n2 P. Bille, I. L. Gørtz, M. Pérez-López, S. R. Tarnow  \nIn what follows, we use the notation nW and mW for the number of vertices and edges of an automaton W , respectively, and define |W| = nW +mW . Alanko et al. describe, in their seminal papers [3, 4], a deterministic Wheeler automaton (WDFA) with the same language of any non-deterministic Wheeler automaton (WNFA), dubbed the Wheeler determinization of the WNFA. It has at most 2nA − 1 − σ nodes and O (nA σ) edges, where σ is the size of the alphabet. They give an O (n3A) time algorithm to construct the Wheeler determinization, which is not output-sensitive given the stated size. Still, this result is remarkable, given that in general graphs the blow-up in size when going from NFAs to DFAs is exponential in the worst case.  \nOur contribution. We describe an algorithm for building the WDFA described by Alanko et al. that runs in output-sensitive time. More precisely:  \nTheorem 1 . Given an input WNFA A and its Wheeler order, there is an algorithm to construct its Wheeler determinization D in O (|A| + |D|) time.  \nGiven that mW = O (nW σ) holds for any Wheeler graph [16], we have a linear time algorithm for sorted inputs. If all vertices of A are reachable from the source (a standard assumption, se","cbCaivXEJQdcD0PH","https://ap.wps.com/l/cbCaivXEJQdcD0PH","pdf",372791,2,1,9,"English","en",105,"# Introduction\n## Wheeler graphs and related applications\n# Contribution and main result\n## Output-sensitive determinization algorithm\n# Preliminaries\n## Automata and graph model","[{\"question\":\"What problem does the paper address in Wheeler determinization?\",\"answer\":\"Given a Wheeler NFA A, the goal is to construct a Wheeler DFA D that accepts exactly the same language as A.\"},{\"question\":\"How does the proposed running time improve over the prior approach?\",\"answer\":\"Using the Wheeler order of A, the algorithm runs in output-sensitive time O(nA + mA + nD + mD), improving on the previously described O(nA^3) time solution.\"},{\"question\":\"Under what conditions does the paper achieve linear time, and why is the bound tight?\",\"answer\":\"For alphabet size σ = O(1), the paper states the first linear-time algorithm. It also shows tightness by constructing a family of WNFA inputs where the resulting determinized DFA is minimum and has worst-case size Θ(nσ).\"}]",1784188814,23,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":28},"tighter-bounds-for-wheeler-determinization","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/tighter-bounds-for-wheeler-determinization/83558/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-23","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 in Wheeler determinization?","Question",{"text":75,"@type":76},"Given a Wheeler NFA A, the goal is to construct a Wheeler DFA D that accepts exactly the same language as A.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the proposed running time improve over the prior approach?",{"text":80,"@type":76},"Using the Wheeler order of A, the algorithm runs in output-sensitive time O(nA + mA + nD + mD), improving on the previously described O(nA^3) time solution.",{"name":82,"@type":73,"acceptedAnswer":83},"Under what conditions does the paper achieve linear time, and why is the bound tight?",{"text":84,"@type":76},"For alphabet size σ = O(1), the paper states the first linear-time algorithm. 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