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 planar graph of size [Formula: see text] 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 (for planar graphs, Borradaile et al. very recently presented a polynomial time approximation scheme) 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).
Abstract. The novel octilinear routing paradigm (X-architecture) in VLSI design requires new approaches for the construction of Steiner trees. In this paper, we consider two versions of the shortest octilinear Steiner tree problem for a given point set K of terminals in the plane: (1) a version in the presence of hard octilinear obstacles, and (2) a version with rectangular soft obstacles. The interior of hard obstacles has to be avoided completely by the Steiner tree. In contrast, the Steiner tree is allowed to run over soft obstacles. But if the Steiner tree intersects some soft obstacle, then no connected component of the induced subtree may be longer than a given fixed length L. This kind of length restriction is motivated by its application in VLSI design where a large Steiner tree requires the insertion of buffers (or inverters) which must not be placed on top of obstacles. For both problem types, we provide reductions to the Steiner tree problem in graphs of polynomial size with the following approximation guarantees. Our main results are (1) a 2-approximation of the octilinear Steiner tree problem in the presence of hard rectilinear or octilinear obstacles which can be computed in O(n log 2 n) time, where n denotes the number of obstacle vertices plus the number of terminals, (2) a (2 + ε)-approximation of the octilinear Steiner tree problem in the presence of soft rectangular obstacles which runs in O(n 3 ) time, and (3) a polynomial time (1.55 + ε)-approximation of the octilinear Steiner tree problem in the presence of soft rectangular obstacles.
Abstract. 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 socalled 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 NPcompleteness 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 ε 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).
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