[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83312-en":3,"doc-seo-83312-105":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},83312,1374391974585,"Genevieve","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Canonical Join Trees","Rooted join trees of acyclic hypergraphs are studied under a canonical depth-minimization criterion: a join tree is canonical if every node has minimum possible depth among all join trees with the same root. Luo et al. posed how acyclic hypergraphs behave with respect to admitting canonical join trees depending on whether none, some, or all hyperedges can serve as the root. This work resolves the open problem by proving uniqueness of each canonical join tree relative to its root, providing an initial characterization, classifying hypergraphs admitting canonical join trees for none/some/all hyperedges, and introducing a linear-time construction algorithm when such a tree exists.","arXiv :2607 .07992v 1 [ cs .DM] 8 Jul 2026  \nCanonical Join Trees  \nArne Leitert  \nAbstract. A rooted join tree of an acyclic hypergraph is canonical if each of its nodes has minimum possible depth among all join trees with the same root. Luo etal. introduce these trees in [9] and pose the open problem of characterizing acyclic hypergraphs according to whether they admit canonical join trees for none, some, or all hyperedges as root. In this paper, we resolve this question. We show that each canonical join tree is unique with respect to its root and give a first characterisation for such trees. Additionally, we characterise hypergraphs that admit a canonical join tree for none, some, or all their hyperedges as root. Lastly, we present a linear-time algorithm that constructs a canonical join tree whenever one exists.  \n1 Introduction  \nAcyclic hypergraphs are a fundamental structure in database theory and graph theory. Their key characterisation is the ability to arrange their hyperedges into a tree such that, if two hyperedges share a vertex x, then every hyperedge on the path between them also contains x. Such a tree is called join tree. They provide a compact structural representation of acyclic hypergraphs and form the basis of many algorithms on them. Recently, join trees have also attracted renewed attention in the database community as tool for optimizing join queries (see for example [7, 9, 12] and literature cited in them) .  \nOut of all join trees an acyclic hypergraph has, trees with small height are of special interest, because such trees allow better parallelization and index utilization. In [1], Blair and Peyton present an algorithm which computes a join tree with minimum diameter (and thus minimum height) in O (N ) time where Nis the total input size of a given hypergraph. Note that their algorithm assumes the maximal cliques of a chordal graph as input. One can therefore use it for any acyclic hypergraph without the overhead of creating a corresponding chordal graph. With the same motivation, Luo et al. [9] introduce the stricter notion of a canonical join tree. Such a tree T simultaneously minimises the depth of every individual node u (denoted as depT (u)) instead of just the overall height:  \nDefinition 1 . A join tree T with root r is canonical if depT (u) ≤ depT′ (u) for any other join tree T′ rooted in r and all nodes u.  \nLuo etal. demonstrate that Berge-acyclic hypergraphs possess a unique canonical join tree for each of their hyperedges as root, and that one can find this tree in linear time. They also raise the question of precisely characterising hypergraphs which admit unique canonical join trees for none, some, or all their hyperedges as root, respectively. We answer this question in this paper. Our specific contributions are as follows:  \n2 Arne Leitert  \n– We show that, every canonical join tree is unique if it exists.  \n– We characterize canonical join trees in terms of their hypergraph’s union join graph.  \n– We provide a subgraph characterization for acyclic hypergraphs that do not admit any canonical join trees.  \n– We present a linear-time algorithm which finds a canonical join tree for a given hypergraph and root, provided one exists.  \n– We characterise hypergraphs that admit a canonical join tree for every hyperedge as root.  \n2 Preliminaries  \nLet H = (V, E) be a hypergraph. We use N = PE∈E |E| to denote the total size of all hyperedges of H. Whenever a hypergraph is given, the input size isin Θ (N ) .  \nThis paper primarily investigates graphs and trees constructed from the hyperedges of some hypergraph. We therefore use the term vertex exclusively for elements with in a hyperedge, and we use the term node for hyperedges in context of such a graph. That is, a node and hyperedge are interchangeable terms in this paper. We use slanted lowercase letters (e. g. x, y) to represent vertices and italic lowercase letters (e. g. u, v) to identify nodes. Keep in mind, however, that nodes are sets ","cbCaib4nBDgYiLYw","https://ap.wps.com/l/cbCaib4nBDgYiLYw","pdf",453165,1,13,"English","en",105,"# Introduction\n## Join trees for acyclic hypergraphs\n## Canonical join tree definition and prior work\n# Preliminaries\n## Notation and graph/tree concepts\n## Join tree and acyclicity\n# Canonical Join Trees","[{\"question\":\"What makes a join tree “canonical” in this paper?\",\"answer\":\"A rooted join tree is canonical if, for every node, its depth is less than or equal to the depth of the same node in any other join tree with the same root.\"},{\"question\":\"What open problem about canonical join trees is resolved?\",\"answer\":\"The paper characterizes acyclic hypergraphs according to whether they admit canonical join trees when none, some, or all hyperedges are chosen as the root.\"},{\"question\":\"Is there an efficient algorithm to construct a canonical join tree?\",\"answer\":\"Yes. The paper presents a linear-time algorithm that constructs a canonical join tree whenever one exists for a given hypergraph and root.\"}]",1784186674,33,{"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},"canonical-join-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/canonical-join-trees/83312/",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 makes a join tree “canonical” in this paper?","Question",{"text":75,"@type":76},"A rooted join tree is canonical if, for every node, its depth is less than or equal to the depth of the same node in any other join tree with the same root.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What open problem about canonical join trees is resolved?",{"text":80,"@type":76},"The paper characterizes acyclic hypergraphs according to whether they admit canonical join trees when none, some, or all hyperedges are chosen as the root.",{"name":82,"@type":73,"acceptedAnswer":83},"Is there an efficient algorithm to construct a canonical join tree?",{"text":84,"@type":76},"Yes. 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