The Steiner Minimal Tree problem in the λ-geometry plane

Abstract

A Steiner Minimal Tree (SMT) for a given set P of points is a shortest network interconnecting the points of P whose vertex set may include some additional points in order to get the minimum possible total length in a metric space. When no additional points are allowed the minimum interconnection network is the well-known minimum spanning tree (MST) of P. The Steiner ratio is the greatest lower bound of the ratio of the length of an SMT over that of an MST of P. In this paper we study the Steiner minimal tree problem in which all the edges of SMT have fixed orientations. We call it the SMT problem in the λ-geometry plane, where λ is the number of possible orientations.

Here is the summary of our results.

• 1. We show that the Steiner ratio for ¦P¦≥3 is √3/2 cos(π/2λ), for λ=6m+3 and integer m≥0, and is √3/2, for λ=6k and integer k≥1, disproving a a conjecture of Du et al.[3] that the ratio is √3/2 iff the unit disk in normed planes is an ellipse.

• 2. We derive the Steiner ratios for ¦P¦ ≤ 4 for all possible λ's and show that for ¦P¦≥3 there exists an SMT whose Steiner points lie in a multi-level Hanan-grid, generalizing a result that holds for rectilinear case, i.e., λ=2.

These results show that the Steiner ratio is not a monotonically increasing function of λ, as believed by many researchers. We conjecture that the Steiner ratios obtained above (¦P¦≤4) are actually true for all ¦P¦≥3.

Supported in part by the National Science Foundation under the Grant CCR-9309743, and by the Office of Naval Research under the Grant No. N00014-93-1-0272.