Approximation of Octilinear Steiner Trees Constrained by Hard and Soft Obstacles

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 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) time, and (3) a polynomial time (1.55 + ε)– approximation of the octilinear Steiner tree problem in the presence of soft rectangular obstacles.