Locally minimal uniformly oriented shortest networks

Abstract

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 generalization 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 [M. Brazil, D.A. Thomas, J.F. Weng, Forbidden subpaths for Steiner minimum networks in uniform orientation metrics, Networks 39 (2002) 186–222]. 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 [M. Brazil, D.A. Thomas, J.F. Weng, Forbidden subpaths for Steiner minimum networks in uniform orientation metrics, Networks 39 (2002) 186–222].