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Can shortest paths among h obstacles in the plane, with a total of n vertices, be found in optimal O(n + h \log h) time using O(n) space?
Uncertain, pending investigation.
Open.
The only algorithm that is linear in n in time and space is quadratic in h [KMM97]; O(n\log n) time, using O(n\log n) space, is known [HS99]. In three dimensions, the Euclidean shortest path problem among general obstacles is NP-hard, but its complexity remains open for some special cases, such as when the obstacles are disjoint unit spheres or axis-aligned boxes; see [Mit00] for a survey.
shortest paths
J. O’Rourke, 2 Aug. 2001.
John Hershberger and Subhash Suri. An optimal algorithm for Euclidean shortest paths in the plane. SIAM J. Comput., 28(6):2215–2256, 1999.
S. Kapoor, S. N. Maheshwari, and Joseph S. B. Mitchell. An efficient algorithm for Euclidean shortest paths among polygonal obstacles in the plane. Discrete Comput. Geom., 18:377–383, 1997.
Joseph S. B. Mitchell. Geometric shortest paths and network optimization. In Jörg-Rüdiger Sack and Jorge Urrutia, editors, Handbook of Computational Geometry, pages 633–701. Elsevier Publishers B.V. North-Holland, Amsterdam, 2000.
J. S. B. Mitchell and Joseph O’Rourke. Computational geometry column 42. Internat. J. Comput. Geom. Appl., 11(5):573–582, 2001. Also in SIGACT News 32(3):63-72 (2001), Issue 120.