Problem 21: Shortest Paths among Obstacles in 2D

Statement

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?

Origin

Uncertain, pending investigation.

Status/Conjectures

Open.

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.

Appearances

[MO01]

Categories

shortest paths

Entry Revision History

J. O'Rourke, 2 Aug. 2001.

Bibliography

HS99

John Hershberger and Subhash Suri.
An optimal algorithm for Euclidean shortest paths in the plane.
SIAM J. Comput., 28(6):2215-2256, 1999.

KMM97

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.

Mit00

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.

MO01

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.