Locally minimal uniformly oriented shortest networks

The Steiner problem in a λ-plane is the problem of constructing a minimum length network interconnecting a given set of nodes (called terminals), with the constraint that all line segments in the network have slopes chosen from λ uniform orientations in the plane. This network is referred to as a minimum λ-tree. The problem is a generalisation of the classical Euclidean and rectilinear Steiner tree problems, with important applications to VLSI wiring design. A λ-tree is said to be locally minimal if its length cannot be reduced by small perturbations of its Steiner points. In this paper we prove that a λ-tree is locally minimal if and only if the length of each path in the tree cannot be reduced under a special parallel perturbation on paths known as a shift. This proves a conjecture on necessary and sufficient conditions for locally minimal λ-trees raised in [2]. For any path P in a λ-tree T , we then find a simple condition, based on the sum of all angles on one side of P , to determine whether a shift on P reduces, preserves, or increases the length of T. This result improves on our previous forbidden paths results in [2].