The Steiner ratio of L2kd

Dietmar Cieslik
1999 Discrete Applied Mathematics  
Let L d 2k be the d-dimensional space with 2k-norm. Given a ÿnite set N of points in this space. Find a connected graph G = (V; E) such that N ⊆ V and the total length of G is minimal. Such a network is called a Steiner minimal tree (SMT). If we connect pairs of given points only, we ÿnd a minimum spanning tree (MST). Whereas an MST is easy to ÿnd a method to construct an SMT in L d 2k needs exponential time for d = 2 and is still unknown for d ¿ 2. The Steiner ratio m(d; 2k) of L d 2k is a
more » ... ure of how good an MST approximates an SMT. We estimate this quantity. ? 0166-218X/99/$ -see front matter ? 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 6 -2 1 8 X ( 9 9 ) 0 0 0 7 6 -1
doi:10.1016/s0166-218x(99)00076-1 fatcat:hlnsagtofbds3pbu4olwcv5424