[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81869-en":3,"doc-seo-81869-105":31,"detail-sidebar-cat-0-en-105":92},{"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},81869,2336464648322,"Aria","https://ap-avatar.wpscdn.com/avatar/2200025388227c56fec?_k=1778556882303663488",8,"Research & Report","Shortest Path Map Equivalence Decompositions and Applications","Given a polygonal domain P in the plane and a source point s, the shortest path map SPM(s) decomposes P into cells where all points t share the same shortest-path vertex sequence. The SPM-equivalence decomposition refines this by making SPM(s) topologically equivalent for all s within one cell. The paper derives new upper bounds for SPM-equivalence decomposition complexities under boundary-restricted variants, and introduces algorithms to compute them. Applications include efficient two-point shortest-path queries, and computing geodesic diameter and center of P.","arXiv :2607 .034 16v 1 [ cs .CG] 3 Jul 2026  \nShortest Path Map Equivalence Decompositions and Applications ∗  \nHaitao Wang†  \nAbstract  \nGiven a polygonal domain P in the plane, the shortest path map with respect to a point s, denoted by SPM (s), is the decomposition of P into cells such that shortest paths from s to all points t in the same cell have the same vertex sequence. The shortest path map equivalence decomposition of P is the decomposition of P into cells so that SPM (s) is topologically equivalent for all points s in the same cell. In this paper, we prove new upper bounds on the combinatorial complexities of the SPM-equivalence decompositions under various settings, depending on whether s and/or t are restricted to the boundary of P. We also propose new algorithms to compute these decompositions. Further, our results lead to new solutions to several other problems, including answering two-point shortest path queries in P, and computing geodesic diameter and center of P.  \nKeywords: shortest paths, shortest path maps, SPM-equivalent decompositions, geodesic diameter, geodesic center, polygons  \n1 Introduction  \nLet P be a polygonal domain of h holes with a total of n vertices in the plane, i.e. , P is a closed and multiply-connected region bounded by n segments that form h+1 cycles (holes and the region outside P are also called obstacles) . The shortest path map with respect to a point s ∈ P, denoted by SPM(s), is the decomposition of P into cells such that shortest paths from s to all points t in the same cell are combinatorially the same (i.e., they have the same vertex sequence) [13, 26]; see Fig. 1. The shortest path map equivalence decomposition (or SPM-equivalence decomposition for short) of P, denoted by Ψ, is the decomposition of P into cells so that SPM (s) is topologically equivalent for all points s in the same cell. Chiang and Mitchell [13] first studied the SPM-equivalence decomposition and used it to answer two-point shortest path queries in P. They proved that the combinatorial complexity of Ψ is bounded by O(n10 ) and proposed an O (n10 log n) time algorithm to compute it.  \nWe consider several variants of the SPM-equivalence decompositions, depending on whether sand/or t are required to be on the boundary of P. Specifically, let B denote the boundary of P. We define the boundary-restricted shortest path map for a point s, denoted by SPMB(s), as the decomposition of B into segments such that shortest paths from s to all points t in the same segment are combinatorially the same. We define the boundary-restricted SPM-equivalence decomposition with respect to P, denoted by ΨB (P), as the decomposition of B into segments so that SPM(s) are topologically equivalent for all points s in the same segment. Similarly, we let ΨB (B) denote the decomposition of B into segments so that SPMB(s) is topologically the same for all points s in the same segment, and ΨP(B) the decomposition of P into cells so that SPMB(s) is topologically the same for all points s in the same cell. If we follow the same convention for notation, then ΨP (P) is Ψ (we will use the two notations interchangeably) . Another way to view these four decompositions is to define them depending on whether s and/or t are restricted to be on B. For example, ΨP (B) is for the case wheres ∈ P while t ∈ B. Note that ΨP(P) is a refinement of ΨP (B) (i.e., each cell of ΨP (P) is completely inside a cell of ΨP(B)), and ΨB (P) is a refinement of ΨB (B) .  \n∗ A preliminary version of this paper will appear in Proceedings of the 34th Annual European Symposium on Algorithms (ESA 2026) .  \n†Kahlert School of Computing, University of Utah, Salt Lake City, UT 84112, [USA.](USA. haitao.wang@utah.edu)[ haitao.wang@utah.edu](USA. haitao.wang@utah.edu)  \nFigure 1: Illustrating a shortest path map SPM (s): The dotted blue segments are extension segments and the dashed red curves are bisector curves. The point t is in a cell whose root is v.  \nWhile we are not aware o","cbCaioJEcXK8I1Po","https://ap.wps.com/l/cbCaioJEcXK8I1Po","pdf",767256,4,1,33,"English","en",105,"# Introduction\n## Shortest path maps and equivalence decompositions\n## Boundary-restricted variants and refinements\n## Summary of new bounds and algorithms","[{\"question\":\"What is a shortest path map (SPM(s)) for a point s in a polygonal domain P?\",\"answer\":\"SPM(s) decomposes P into cells such that for any two points t in the same cell, the shortest paths from s to t have the same vertex sequence (combinatorially the same).\"},{\"question\":\"How is the SPM-equivalence decomposition defined?\",\"answer\":\"The SPM-equivalence decomposition splits P into cells so that for any two source points s in the same cell, the shortest path maps SPM(s) are topologically equivalent.\"},{\"question\":\"What problem areas do the paper’s results address beyond constructing these decompositions?\",\"answer\":\"The results enable new solutions for two-point shortest path queries in P, and for computing the geodesic diameter and geodesic center of P.\"}]","Shortest Path Map Equivalence Decompositions and Applications | 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is a shortest path map (SPM(s)) for a point s in a polygonal domain P?","Question",{"text":76,"@type":77},"SPM(s) decomposes P into cells such that for any two points t in the same cell, the shortest paths from s to t have the same vertex sequence (combinatorially the same).","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How is the SPM-equivalence decomposition defined?",{"text":81,"@type":77},"The SPM-equivalence decomposition splits P into cells so that for any two source points s in the same cell, the shortest path maps SPM(s) are topologically equivalent.",{"name":83,"@type":74,"acceptedAnswer":84},"What problem areas do the paper’s results address beyond constructing these decompositions?",{"text":85,"@type":77},"The results enable new solutions for two-point shortest path queries in P, and for computing the geodesic diameter and geodesic center of 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