We show that a metric space embeds in the rectilinear plane (i.e., is L 1-embeddable in R 2) if and only if every subspace with five or six points does. A simple construction shows that for higher dimensions k of the host rectilinear space the number c(k) of points that need to be tested grows at least quadratically with k, thus disproving a conjecture of Seth and Jerome Malitz.

In the rectilinear Steiner arborescence problem the task is to build a shortest rectilinear Steiner tree connecting a given root and a set of terminals which are placed in the plane such that all root-terminal-paths are shortest paths. This problem is known to be NP-hard. In this paper we consider a more restricted version of this problem. In our case we have a depth restrictions d(t) ∈ N for e...

The planar rectilinear Steiner tree problem has been extensively studied. The common formulation ignores circuit fabrication issues such as multiple routing layers, preferred routing directions, and vias between layers. In this paper, the authors extend a previously presented planar rectilinear Steiner tree heuristic to consider layer assignment, preferred routing direction restrictions, and vi...

The functioning of the first type, the flue or labial pipe, was already investigated within previous extensive research projects. The second pipe type, the reed pipe, is less often installed in pipe organs, and only little research was done in this field so far. This is why this pipe type is now the focus of research of the working group ‚Musical Acoustics ‘ at the Fraunhofer Institute for Buil...

We extend known results regarding the maximum rectilinear crossing number of the cycle graph (Cn) and the complete graph (Kn) to the class of general d-regular graphs Rn,d. We present the generalized star drawings of the d-regular graphs Sn,d of order n where n + d ≡ 1 (mod 2) and prove that they maximize the maximum rectilinear crossing numbers. A star-like drawing of Sn,d for n ≡ d ≡ 0 (mod 2...

The spanning ratio and maximum detour of a graph G embedded in a metric space measure how well G approximates the minimum complete graph containing G and metric space, respectively. In this paper we show that computing the spanning ratio of a rectilinear path P in L1 space has a lower bound of Ω(n log n) in the algebraic computation tree model and describe a deterministic O(n log n) time algori...

We consider the following problem known as simultaneous geometric graph embedding (SGE). Given a set of planar graphs on a shared vertex set, decide whether the vertices can be placed in the plane in such a way that for each graph the straight-line drawing is planar. We partially settle an open problem of Erten and Kobourov [5] by showing that even for two graphs the problem is NP-hard. We also...