Reducing the Steiner problem in a normal space
SIAM Journal on Computing
Approximating geometrical graphs via “spanners” and “banyans”
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
The lambda-geometry Steiner minimal tree problem and visualization
The lambda-geometry Steiner minimal tree problem and visualization
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
The X architecture: not your father's diagonal wiring
SLIP '02 Proceedings of the 2002 international workshop on System-level interconnect prediction
Minimum Networks in Uniform Orientation Metrics
SIAM Journal on Computing
The Steiner Minimal Tree Problem in the lambda-Geormetry Plane
ISAAC '96 Proceedings of the 7th International Symposium on Algorithms and Computation
An Exact Algorithm for the Uniformly-Oriented Steiner Tree Problem
ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
Constructing exact octagonal steiner minimal trees
Proceedings of the 13th ACM Great Lakes symposium on VLSI
A new paradigm for general architecture routing
Proceedings of the 14th ACM Great Lakes symposium on VLSI
Efficient octilinear Steiner tree construction based on spanning graphs
Proceedings of the 2004 Asia and South Pacific Design Automation Conference
Highly scalable algorithms for rectilinear and octilinear Steiner trees
ASP-DAC '03 Proceedings of the 2003 Asia and South Pacific Design Automation Conference
Approximation of octilinear steiner trees constrained by hard and soft obstacles
SWAT'06 Proceedings of the 10th Scandinavian conference on Algorithm Theory
A near linear time approximation scheme for steiner tree among obstacles in the plane
WADS'07 Proceedings of the 10th international conference on Algorithms and Data Structures
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Given a point set K of terminals in the plane, the octilinear Steiner tree problem is to find a shortest tree that interconnects all terminals and edges run either in horizontal, vertical, or ± 45° diagonal direction. This problem is fundamental for the novel octilinear routing paradigm in VLSI design, the so-called X-architecture. As the related rectilinear and the Euclidian Steiner tree problem are well-known to be NP-hard, the same was widely believed for the octilinear Steiner tree problem but left open for quite some time. In this paper, we prove the NP-completeness of the decision version of the octilinear Steiner tree problem. We also show how to reduce the octilinear Steiner tree problem to the Steiner tree problem in graphs of polynomial size with the following approximation guarantee. We construct a graph of size $O({{n^{2}}\over{\varepsilon^{2}}})$ which contains a (1+ε)–approximation of a minimum octilinear Steiner tree for every ε 0 and n = |K|. Hence, we can apply any α-approximation algorithm for the Steiner tree problem in graphs (the currently best known bound is α ≈ 1.55) and achieve an (α+ε)- approximation bound for the octilinear Steiner tree problem. This approximation guarantee also holds for the more difficult case where the Steiner tree has to avoid blockages (obstacles bounded by octilinear polygons).