[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84040-en":3,"doc-seo-84040-105":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},84040,13056703019404,"Miles","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","The Surplus Parking Gathering Problem in Infinite Grids","The Surplus Parking Gathering Problem (SPG) coordinates autonomous robots on an infinite grid with designated parking nodes, each having a fixed capacity, while the robot population exceeds total parking capacity. The task saturates every parking node exactly to its capacity and gathers all remaining surplus robots at one common grid node whose location is not given in advance. Robots operate under an asynchronous model with global visibility and global strong multiplicity detection. The paper derives necessary solvability conditions, proposes a collision-free deterministic distributed algorithm for all solvable cases, proves finite termination with uniquely determined gathering, and bounds move complexity by O(n(a+b)+n^2) and Ω(n(a+b)).","arXiv :2607 .05983v 1 [ cs .DC] 7 Jul 2026  \nThe Surplus Parking Gathering Problem in Infinite  \nGrids  \nAnimesh Maiti 1 Abhinav Chakraborty2 Subhash Bhagat 1  \n1 Department of Mathematics  \nIndian Institute of Technology Jodhpur, Rajasthan, India  \nEmail: [animesh7.iitj@gmail.com](animesh7.iitj@gmail.com) , [sbhagat@iitj.ac.in](sbhagat@iitj.ac.in)  \n2 Department of Mathematics  \nBirla Institute of Technology Mesra, Ranchi, Jharkhand, India  \nEmail: [abhinavchakraborty@bitmesra.ac.in](abhinavchakraborty@bitmesra.ac.in)  \nAbstract  \nIn this paper, we introduce the Surplus Parking Gathering Problem (SPG), a new coordination problem for robots deployed on an infinite grid. The input consists of a set of designated parking nodes, each associated with a prescribed capacity, while the total number of robots exceeds the total parking capacity. The objective is to saturate every parking node exactly according to its capacity while gathering all remaining surplus robots at a common grid node that is not specified a priori. The robots are assumed to be autonomous, anonymous, oblivious, identical, disoriented, and homogeneous. We consider the asynchronous (async) model with global visibility and global strong multiplicity detection.  \nWe first establish necessary conditions for the solvability of SPG by characterizing the initial configurations that admit no deterministic distributed algorithm. For all the remaining solvable configurations, we present a deterministic distributed algorithm that correctly solves the problem. The proposed algorithm proceeds in several phases and avoids collisions throughout its execution. We prove that the algorithm terminates in finite time and, upon termination, every parking node is saturated according to its prescribed capacity while all surplus robots are gathered at a uniquely determined gathering node. We further analyze the move complexity of the proposed algorithm, obtaining an O (n(a + b) + n2 ) upper bound together with an Ω(n(a + b)) worst-case lower bound for the SPG problem.  \n1 Introduction  \nThe coordination of large collections of autonomous mobile robots has been one of the central research topics in distributed computing and swarm robotics over the past three decades. Fundamental tasks such as gathering [13, 31, 14, 19], pattern formation [21, 7, 10], exploration [20, 3] and mutual visibility [24, 1, 26] have been extensively studied under various computational models. These problems are motivated by various applications including automated parking systems, environmental monitoring, search-and-rescue operations, and precision agriculture [29, 2] . The primary objective is to design distributed algorithms that enable robots to accomplish a common task despite their limited computational capabilities and the absence of explicit communication.  \nIn the traditional robot model, the robots are assumed to be autonomous (there is no central co-  \nordinator), anonymous (they have no unique identifiers), homogeneous (all robots execute the same deterministic algorithm), and identical (they are physically indistinguishable) . Depending on the underlying computational model, robots may operate either in a continuous Euclidean space or on the nodes of a graph. Robots are generally modeled as dimensionless points in Euclidean space in most existing work. In recent years, graph-based models have attracted considerable attention because they are able to capture many structured environments where robot movements are constrained by the topology of the underlying network. In this work, we consider robots deployed on the nodes of an infinite grid, where robot movements are restricted to adjacent grid nodes [4, 17] .  \nThe robots do not share a common global coordinate system; instead, each robot is equipped with its own local coordinate system whose origin is located at its current position. The directions and orientations of the coordinate axes may differ from one robot to another. However, since the robot","cbCaihFilRlxetEg","https://ap.wps.com/l/cbCaihFilRlxetEg","pdf",3881161,3,1,50,"English","en",105,"# Introduction\n## Robot and Grid Model\n## Problem Definition: SPG\n## Solvability Conditions\n## Deterministic Distributed Algorithm\n## Termination and Correctness\n## Move Complexity","[{\"question\":\"What does the Surplus Parking Gathering Problem require robots to achieve?\",\"answer\":\"SPG requires saturating each designated parking node exactly according to its prescribed capacity, while gathering all remaining surplus robots at a single common grid node.\"},{\"question\":\"Why is the gathering node location considered part of the challenge?\",\"answer\":\"The surplus robots must be gathered at a common node whose location is not specified a priori, so the algorithm must determine it through robot coordination.\"},{\"question\":\"Under what robot assumptions and computational model does the paper work?\",\"answer\":\"Robots are autonomous, anonymous, oblivious, identical, disoriented, and homogeneous. The solution is designed for an asynchronous model with global visibility and global strong multiplicity detection.\"}]",1784192185,126,{"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},"the-surplus-parking-gathering-problem-in-infinite-grids","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/the-surplus-parking-gathering-problem-in-infinite-grids/84040/",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","2026-07-27","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 does the Surplus Parking Gathering Problem require robots to achieve?","Question",{"text":75,"@type":76},"SPG requires saturating each designated parking node exactly according to its prescribed capacity, while gathering all remaining surplus robots at a single common grid node.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why is the gathering node location considered part of the challenge?",{"text":80,"@type":76},"The surplus robots must be gathered at a common node whose location is not specified a priori, so the algorithm must determine it through robot coordination.",{"name":82,"@type":73,"acceptedAnswer":83},"Under what robot assumptions and computational model does the paper work?",{"text":84,"@type":76},"Robots are autonomous, anonymous, oblivious, identical, disoriented, and homogeneous. 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