Approximation algorithms for shortest path motion planning
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
An approximation scheme for finding Steiner trees with obstacles
SIAM Journal on Computing
The steiner problem with edge lengths 1 and 2,
Information Processing Letters
Approximating geometrical graphs via “spanners” and “banyans”
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems
Journal of the ACM (JACM)
Improved Steiner tree approximation in graphs
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
Minimum Networks in Uniform Orientation Metrics
SIAM Journal on Computing
Obstacle-Avoiding Euclidean Steiner Trees in the Plane: An Exact Algorithm
ALENEX '99 Selected papers from the International Workshop on Algorithm Engineering and Experimentation
Planar Spanners and Approximate Shortest Path Queries among Obstacles in the Plane
ESA '96 Proceedings of the Fourth Annual European Symposium on Algorithms
I/o-efficient algorithms for shortest path related problems
I/o-efficient algorithms for shortest path related problems
A polynomial-time approximation scheme for Steiner tree in planar graphs
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Approximation of octilinear steiner trees constrained by hard and soft obstacles
SWAT'06 Proceedings of the 10th Scandinavian conference on Algorithm Theory
Hardness and approximation of octilinear steiner trees
ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
WADS'07 Proceedings of the 10th international conference on Algorithms and Data Structures
Rectilinear paths with minimum segment lengths
Discrete Applied Mathematics
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We present a polynomial time approximation scheme (PTAS) for the Steiner tree problem with polygonal obstacles in the plane with running time O(n log2 n), where n denotes the number of terminals plus obstacle vertices. To this end, we show how a planar spanner of size O(n log n) can be constructed that contains a (1+ε)-approximation of the optimal tree. Then one can find an approximately optimal Steiner tree in the spanner using the algorithm of Borradaile et al. (2007) for the Steiner tree problem in planar graphs.We prove this result for the Euclidean metric and also for all uniform orientation metrics, i.e. particularly the rectilinear and octilinear metrics.