Steiner Minimum Trees for Equidistant Points on Two Sides of an Angle

  Rainer E. Burkard Tibor Dudás

Abstract:

  In this paper we deal with the Steiner minimum tree problem for a special type of point sets. These sets consist of the vertex of an angle and equidistant points lying on the two sides of this angle.

 Jarník and Kössler (1934) formulated the following problem: Determine the shortest tree which connects given points in the plane. Seven years later, Courant and Robbins (1941) describe this problem in their classical book ``What is Mathematics?" and contribute this problem for to J. Steiner, though Torricelli and Cavalieri gave solutions for the triangle already in 1640. For an account on the history of this problem see Hwang, Richards and Winter (1992). Since Courant and Robbins this problem is called Steiner Minimum Tree (SMT) Problem .

 For an arbitrary point set in the plane with the problem is quite difficult. Until Melzak (1961) it was not even known that it is finitely solvable. Garey, Graham and Johnson (1977) proved that the Steiner minimum tree problem is -hard. This means that unless there does not exist a polynomial (and efficient) algorithm which solves this problem. Therefore a considerable interest arose in studying special point sets for which an SMT can be found in polynomial time. The first special point sets considered were ladders , see Chung and Graham (1978) and the recent correction by Burkard, Dudás and Maier (1994). Other special point sets include zigzag lines (Du, Hwang and Weng, 1983), checkerboards (Chung, Gardner and Graham, 1989), Chinese checkerboards (Hwang and Du, 1991), bar waves (Du and Hwang, 1987), sets of four points (Du, Hwang, Song and Weng), regular polygons (Du, Hwang and Weng, 1987) and points on a circle (Du, Hwang and Chao, 1985).

 In this article we contribute a new special case : triangle ladders , where the given points consist of the vertex of an angle and further points lying equidistantly on the two sides of this angle. We shall determine the structure of an SMT in dependence of the angle .

 Gyenizse Pal 1997. Február 4. Kedd 14:22:04 MET