[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81899-en":3,"doc-seo-81899-105":31,"detail-sidebar-cat-0-en-105":85},{"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":28,"seo_description":14,"update_tm":29,"read_time":30},81899,8796095462418,"Noah","https://ap-avatar.wpscdn.com/avatar/80000253c1241d02b47?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778826106357471780",8,"Research & Report","An Exact Generalized k-Cell Decomposition","This paper presents an exact k-cell decomposition for visibility planning in 2D polygonal environments with k-modems, where agents can see through up to k walls. The method guarantees that key visibility events—appear, disappear, split, and merge—occur on every line of the decomposition, removing redundant partition lines found in prior approaches. As a result, the approach achieves O(n^4) complexity and extends to polygons with holes. Applications include optimal pursuit-evasion under k-visibility and counting agents in invisible regions.","An Exact Generalized k-Cell Decomposition  \nYeganeh Bahoo† Sajad Saeedi∗ Roni Sherman†  \narXiv :2607 .0456 1v 1 [ cs .CG] 6 Jul 2026  \nAbstract  \nThis paper introduces an exact k-cell decomposition for visibility planning in polygonal environments for agents equipped with k-modems, devices that can see through up to k walls. Unlike prior decompositions that may include redundant partition lines, our proposed method ensures that visibility events (appear, disappear, merge, and split) are guaranteed to occur on every line of the decomposition. By eliminating these redundancies, we achieve an O(n4 ) complexity , representing a potentially quadratic improvement over the previous best O (k2 n4 ) result. This decomposition explicitly identifies the locations of all critical visibility events and extends to polygons with holes. It has practical applications in tasks such as optimal pursuit-evasion under k-visibility and agent counting in invisible regions.  \n1 Introduction  \nThe ability to detect and capture intruders in constrained environments is a fundamental challenge in areas such as surveillance, search and rescue, autonomous robotics, and wireless sensor networks [14, 9] . In these applications, it is critical to develop strategies that ensure complete exploration of a space, particularly when visibility is limited by structural barriers such as walls. Traditional pursuit-evasion scenarios assume direct lineof-sight between a searcher and an intruder, but in many practical settings, devices or agents are equipped with sensing capabilities that allow them to see through a limited number of obstacles. Such devices are referred to as k-modems [10] . This generalization introduces the concept of k-visibility, that is, the ability to see through k walls, enabling agents to detect intruders even when direct visibility is not feasible. Designing reliable motion strategies under k-visibility is essential for improving situational awareness and guaranteeing successful detection in partially obstructed environments.  \nDespite its importance, pursuit-evasion under kvisibility introduces several geometric and algorithmic challenges [13, 12] . As a pursuer equipped with a k-  \n∗ University College London, London, UK, [s.saeedi@ucl.ac.uk](s.saeedi@ucl.ac.uk)  \n†Toronto Metropolitan University, Toronto, Canada, {bahoo, [roni.sherman](roni.sherman}@torontomu.ca)[}](roni.sherman}@torontomu.ca)[@torontomu.ca](roni.sherman}@torontomu.ca)  \nWe acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC) .  \nmodem moves through a polygonal environment, the visibility region may undergo sudden combinatorial changes, known as geometric events, including appear, disappear, split, and merge [16] . These events complicate the tracking process, as an evader can exploit transient blind spots to avoid detection. Predicting when and where these events occur is computationally nontrivial and requires a deep understanding of visibility dynamics. Moreover, constructing a representation of the environment that guarantees the stability of visibility regions during movement is difficult, particularly ask increases.  \nTo address this, we build upon the concept of visibility-based cell decomposition introduced by Guibas et al. [11], who partitioned polygons to facilitate searching for intruders under 0-visibility. This approach was later extended to 2-visibility by Bahoo et al. [6] and subsequently generalized to k-visibility in [7] . While the decomposition in [7] ensures that the combinatorial representation of the shadow remains invariant within each cell, it is not ”exact” -many of its partition lines do not correspond to actual visibility events, leading to unnecessary overhead. The main contribution of this work is the development of an exact k-visibility cell decomposition for 2D polygonal environments. By identifying and utilizing only the lines where visibility events are guaranteed to occur, we achieve a more sparse ","cbCaidJHRydSqjj4","https://ap.wps.com/l/cbCaidJHRydSqjj4","pdf",858506,5,1,15,"English","en",105,"# Introduction\n## Overview of k-visibility and k-modems\n## Motivation and challenges\n## Related work and main contribution\n# Preliminaries\n## Modeling and key definitions\n# Proposed approach\n## Exact k-cell decomposition\n# Complexity analysis\n# Extension to polygons with holes\n# Conclusions","[{\"question\":\"What are practical applications of the decomposition?\",\"answer\":\"The paper highlights tasks such as optimal pursuit-evasion under k-visibility and counting agents in invisible regions, including for environments with holes.\"}]","An Exact Generalized k-Cell Decomposition | PDF",1784176943,38,{"code":4,"msg":32,"data":33},"ok",{"site_id":25,"language":24,"slug":34,"title":13,"keywords":35,"description":14,"schema_data":36,"social_meta":80,"head_meta":82,"extra_data":84,"updated_unix":29},"an-exact-generalized-k-cell-decomposition","",{"@graph":37,"@context":79},[38,55,70],{"@type":39,"itemListElement":40},"BreadcrumbList",[41,45,49,52],{"item":42,"name":43,"@type":44,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":46,"name":47,"@type":44,"position":48},"https://docshare.wps.com/document/","Document",2,{"item":50,"name":12,"@type":44,"position":51},"https://docshare.wps.com/document/research-report/",3,{"item":53,"name":13,"@type":44,"position":54},"https://docshare.wps.com/document/an-exact-generalized-k-cell-decomposition/81899/",4,{"url":53,"name":13,"@type":56,"author":57,"headline":13,"publisher":59,"fileFormat":62,"inLanguage":24,"description":14,"dateModified":63,"datePublished":64,"encodingFormat":62,"isAccessibleForFree":65,"interactionStatistic":66},"DigitalDocument",{"name":9,"@type":58},"Person",{"url":42,"name":60,"@type":61},"DocShare","Organization","application/pdf","2026-07-29","2026-07-16",true,{"@type":67,"interactionType":68,"userInteractionCount":20},"InteractionCounter",{"@type":69},"ViewAction",{"@type":71,"mainEntity":72},"FAQPage",[73],{"name":74,"@type":75,"acceptedAnswer":76},"What are practical applications of the decomposition?","Question",{"text":77,"@type":78},"The paper highlights tasks such as optimal pursuit-evasion under k-visibility and counting agents in invisible regions, including for environments with holes.","Answer","https://schema.org",{"og:url":53,"og:type":81,"og:title":13,"og:site_name":60,"og:description":14},"article",{"robots":83,"canonical":53},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":86},[87,91,95,99,103,108,113,116,121,124,128],{"id":21,"doc_module":4,"doc_module_name":47,"category_name":88,"show_sort_weight":89,"slug":90},"Story & Novel",90,"story-novel",{"id":48,"doc_module":4,"doc_module_name":47,"category_name":92,"show_sort_weight":93,"slug":94},"Literature",80,"literature",{"id":54,"doc_module":4,"doc_module_name":47,"category_name":96,"show_sort_weight":97,"slug":98},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":47,"category_name":100,"show_sort_weight":101,"slug":102},"Comic",60,"comic",{"id":104,"doc_module":4,"doc_module_name":47,"category_name":105,"show_sort_weight":106,"slug":107},6,"Technology",50,"technology",{"id":109,"doc_module":4,"doc_module_name":47,"category_name":110,"show_sort_weight":111,"slug":112},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":47,"category_name":12,"show_sort_weight":114,"slug":115},30,"research-report",{"id":117,"doc_module":4,"doc_module_name":47,"category_name":118,"show_sort_weight":119,"slug":120},9,"Religion & Spirituality",20,"religion-spirituality",{"id":119,"doc_module":4,"doc_module_name":47,"category_name":122,"show_sort_weight":119,"slug":123},"World Cup","world-cup",{"id":125,"doc_module":4,"doc_module_name":47,"category_name":126,"show_sort_weight":125,"slug":127},10,"Lifestyle","lifestyle",{"id":129,"doc_module":4,"doc_module_name":47,"category_name":130,"show_sort_weight":20,"slug":131},19,"General","general"]