The rectilinear Steiner tree problem with given topology and length restrictions

We consider the problem of embedding the Steiner points of a Steiner tree with given topology into the rectilinear plane. Thereby, the length of the path between a distinguished terminal and each other terminal must not exceed given length restrictions. We want to minimize the total length of the tree. The problem can be formulated as a linear program and therefore it is solvable in polynomial time. In this paper we analyze the structure of feasible embeddings and give a combinatorial polynomial time algorithm for the problem. Our algorithm combines a dynamic programming approach and binary search and relies on the total unimodularity of a matrix appearing in a sub-problem.