Locally minimal uniformly oriented shortest networks

  • Authors:

  • M. Brazil;D. A. Thomas;J. F. Weng

  • Affiliations:

  • ARC Special Research Centre for Ultra-Broadband Information Networks, Department of Electrical and Electronic Engineering, The University of Melbourne, Vict., Australia;ARC Special Research Centre for Ultra-Broadband Information Networks, Department of Electrical and Electronic Engineering, The University of Melbourne, Vict., Australia;ARC Special Research Centre for Ultra-Broadband Information Networks, Department of Electrical and Electronic Engineering, The University of Melbourne, Vict., Australia

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2006

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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].