{"id":159372,"date":"2010-01-01T00:00:00","date_gmt":"2010-01-01T00:00:00","guid":{"rendered":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/msr-research-item\/querying-approximate-shortest-paths-in-anisotropic-regions-2\/"},"modified":"2018-10-16T21:26:33","modified_gmt":"2018-10-17T04:26:33","slug":"querying-approximate-shortest-paths-in-anisotropic-regions-2","status":"publish","type":"msr-research-item","link":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/publication\/querying-approximate-shortest-paths-in-anisotropic-regions-2\/","title":{"rendered":"Querying Approximate Shortest Paths in Anisotropic Regions"},"content":{"rendered":"<div class=\"asset-content\">\n<p>We present a data structure for answering approximate shortest path queries in a planar subdivision from a fixed source. Let 0<=\u03c1<=1 be a real number. Distances in each face of this subdivision are measured by a possibly asymmetric convex distance function whose unit disk is contained in a concentric unit Euclidean disk and contains a concentric Euclidean disk with radius 1\/\u03c1. Different convex distance functions may be used for different faces, and obstacles are allowed.<\/p>\n<p>Let \u03b5 be any number strictly between 0 and 1. Our data structure returns a (1+\u03b5) approximation of the shortest path cost from the fixed source to a query destination in O(log \u03c1n\u03b5 ) time. Afterwards, a (1 +\u03b5)-approximate shortest path can be reported in O(log n) time plus the complexity of the path. The data structure uses O( \u03c12n3\u03b52 log \u03c1n\u03b5 ) space and can be built in O( \u03c12n3\u03b52 (log \u03c1n\u03b5 )2) time. Our time and space bounds do not depend on any other parameter; in particular, they do not depend on any geometric parameter of the subdivision such as the minimum angle.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>We present a data structure for answering approximate shortest path queries in a planar subdivision from a fixed source. Let 0<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":[{"type":"user_nicename","value":"yajunw","user_id":"34953"}],"msr_publishername":"Society for Industrial and Applied Mathematics","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"5","msr_journal":"SIAM J. 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