[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-650978-105":59,"doc-detail-650978-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","drawing-unordered-trees-on-k-grids","Drawing Unordered Trees on k-Grids","","Almost linear area bounds are established for drawing complete trees on the octagonal k-grid. For 7-ary trees, tight upper and lower bounds are given as Θ(n^1.129), and for ternary trees, bounds O(n^1.048) and Θ(n) are proved, where achieving Θ(n) requires edge bends. The paper studies unit edge length and area complexity for unordered trees on k-grids with k ∈ {4, 6, 8}, and generalizes NP-hardness results from orthogonal and hexagonal grids to the octagonal grid.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/drawing-unordered-trees-on-k-grids/650978/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/drawing-unordered-trees-on-k-grids/650978.png","ImageObject",300,407,{"name":92,"@type":93},"Maya Linwood","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-24",true,{"@type":102,"interactionType":103,"userInteractionCount":81},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What area bounds are proved for complete 7-ary trees on the octagonal grid?","Question",{"text":112,"@type":113},"The paper establishes tight almost linear area bounds, giving an upper and lower bound of Θ(n^1.129) for complete 7-ary trees on the octagonal grid.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How are ternary trees drawn on the 8-grid, and when is Θ(n) achieved?",{"text":117,"@type":113},"For ternary trees, the paper proves upper and lower bounds of O(n^1.048) and Θ(n). The Θ(n) bound specifically needs edge bends.",{"name":119,"@type":110,"acceptedAnswer":120},"What complexity result is generalized to the octagonal grid for unordered trees?",{"text":121,"@type":113},"The paper generalizes NP-hardness results known for orthogonal and hexagonal grids to the octagonal grid, including undecidability questions about unit edge length drawings and area-bounded drawings for trees of bounded degree.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},650978,1792855692,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":81,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":129,"read_time":144},962084928432,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Drawing Unordered Trees on k-Grids  \nChristian Bachmaier and Marco Matzeder  \nUniversity of Passau, Germany  \n{bachmaier,matzeder}@fim.uni-passau.de  \nAbstract. We present almost linear area bounds for drawing complete trees on the octagonal grid. For 7-ary trees we establish an upper and lower bound of Θ(n1 . 129 ) and for ternary trees the bounds of O (n1 .048 ) and Θ (n), where the latter needs edge bends. We explore the unit edge length and area complexity of drawing unordered trees on k-grids with k ∈ {4, 6 , 8} and generalize the N P-hardness results of the orthogonal and hexagonal grid to the octagonal grid.  \n1 Introduction  \nTrees are a fundamental data structure in computer science to represent hierarchies. Amongst others they are used as family trees in social networks or inheritance structures in UML-diagrams. Their visualization is an important field in graph drawing [7,9,13,14,21] . Often trees are unordered, e. g., flow charts. Then it is not necessary that a drawing reflects a given child order. For readable and comprehensible drawings in traditional hierarchical style the following aesthetics are established [17,20,22]: y-coordinates of the vertices correspond to their depth, centered parents over their children, minimal distance between vertices, integral coordinates, maintaining the order, planarity, and identically drawn isomorphic subtrees up to reflection. These criteria exclude recursive winding techniques as they were studied by Chan et al. [7] . Marriott and Stuckey [19] have shown that for unordered binary trees, it is NP-hard to determine a hierarchical drawing with minimal width. The same was shown by Supowit and Reingold [22] for order-preserving drawings. The common drawing algorithm for binary trees was introduced by Reingold and Tilford [20] and generalized to d-ary trees by Walker [3, 6, 24], which all satisfy the above aesthetic criteria.  \nThe hierarchical drawing methods enforce placing the vertices at grid points. All these approaches allow drawing trees of high degree, such that the angles between incident edges may be very small. Restricting the degree of trees allows to draw along a finite set of directions, e. g., four directions on the orthogonal grid. This grid was widely investigated in literature [2,7,8,11,13,14,21,23] . The 6-and the 8-grid with additional axes are used to draw trees with higher degree [1,5, 16]. A motivation for such grids are discrete representations of radial drawings [9] . In our companion paper [5] we showed that it is NP-hard to determine the existence of an order-preserving tree drawing within a given area on the k-grid with k ∈ {4, 6 , 8} . Now we translate the N P-hardness to the unordered case. Bhatt and Cosmadakis [2] showed that it is NP-hard to determine if a tree of  \n2 C. Bachmaier, M. Matzeder  \ndegree up to four has an unit edge length drawing on the orthogonal grid. This result can also be proved by the logic engine approach [10] . For binary trees this was shown by Gregori [15] . NP-hardness results for minimum area werepresented in [4, 18] . We presented an equivalent result for the 6-grid for trees with degree up to six [1] . In the plane only two grid axes are linear independent. This shall cause some problems which do not occur on the 4-grid for compacting drawings on higher order grids containing more than 2 axes. Furthermore, the degree of difficulty increases with the number of available directions.  \nThe remainder is organized as follows. After some definitions in Sect. 2 we present a tight area bound for drawing complete 7-ary trees in Sect. 3. Afterwards we show an almost linear upper area bound for straight-line drawings and a linear bound for drawings with bends of ternary trees on the 8-grid in Sect. 4. Finally, we show that it is NP-hard to decide whether or not there is a unit edge length drawing for arbitrary trees with degree 8 and whether or not there is a drawing within a given area in Sect. 5.  \n2 Preliminaries  \nThe orthogonal or 4-g","cbCaitWxAgaU8zI8","https://ap.wps.com/l/cbCaitWxAgaU8zI8","pdf",462667,13,"English","# Introduction\n## Preliminaries\n## Grid definitions and drawing styles\n## Area bounds and complexity results\n# NP-hardness results","[{\"question\":\"What area bounds are proved for complete 7-ary trees on the octagonal grid?\",\"answer\":\"The paper establishes tight almost linear area bounds, giving an upper and lower bound of Θ(n^1.129) for complete 7-ary trees on the octagonal grid.\"},{\"question\":\"How are ternary trees drawn on the 8-grid, and when is Θ(n) achieved?\",\"answer\":\"For ternary trees, the paper proves upper and lower bounds of O(n^1.048) and Θ(n). The Θ(n) bound specifically needs edge bends.\"},{\"question\":\"What complexity result is generalized to the octagonal grid for unordered trees?\",\"answer\":\"The paper generalizes NP-hardness results known for orthogonal and hexagonal grids to the octagonal grid, including undecidability questions about unit edge length drawings and area-bounded drawings for trees of bounded degree.\"}]","Drawing Unordered Trees on k-Grids | PDF",33]