We study the problem of finding shortest paths in the plane among h convex obstacles, where the path is allowed to pass through (violate) up to k obstacles, for k <= h. Equivalently, the problem is to find shortest paths that become obstacle-free if k obstacles are removed from the input. Given a fixed source point s, we show how to construct a map, called a shortest k-path map, so that all destinations in the same region of the map have the same combinatorial shortest path passing through at most k obstacles. We prove a tight bound of Theta(kn) on the size of this map, and show that it can be computed in O(k^2 n log n) time, where n is the total number of obstacle vertices.
@InProceedings{hershberger_et_al:LIPIcs.ESA.2017.49, author = {Hershberger, John and Kumar, Neeraj and Suri, Subhash}, title = {{Shortest Paths in the Plane with Obstacle Violations}}, booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)}, pages = {49:1--49:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-049-1}, ISSN = {1868-8969}, year = {2017}, volume = {87}, editor = {Pruhs, Kirk and Sohler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.49}, URN = {urn:nbn:de:0030-drops-78413}, doi = {10.4230/LIPIcs.ESA.2017.49}, annote = {Keywords: Shortest paths, Polygonal obstacles, Continuous Dijkstra, Obstacle crossing, Visibility} }
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