Minimizing path lengths in rectilinear Steiner minimum trees with fixed topology

Given a Steiner tree G interconnecting a set of points T located in Rd, the Rectilinear Steiner Tree Problem with Fixed Topology is to compute coordinates for the remaining nodes V(G)∖T minimizing the l1-length of G. Considering the secondary objective of minimizing root-to-leave distances, we prove the existence of optimum coordinates such that among all minimal length solutions all the root-to-leave distances are minimized simultaneously, and we present an O(n⋅d) time algorithm.