The Open Problems Project

Next: Problem 22: Minimum-Link Path in 2D

Previous: Problem 20: Minimum Stabbing Spanning Tree

Problem 21: Shortest Paths among Obstacles in 2D


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.



Partial and Related Results

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

Entry Revision History

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.