NODE-WEIGHTED STEINER FOREST is the following problem: Given an undirected graph, a set of pairs of terminal vertices, a weight function on the vertices, find a minimum weight set of vertices that includes and connects each pair of terminals. We consider the restriction to planar graphs where the problem remains NP-complete. Demaine et al. [DHK09] showed that the generic primal-dual algorithm o...

Given a directed graph G = (V,E) and an integer k ≥ 1, a Steiner k-transitive-closure-spanner (Steiner k-TC-spanner) of G is a directed graph H = (VH , EH) such that (1) V ⊆ VH and (2) for all vertices v, u ∈ V , the distance from v to u in H is at most k if u is reachable from v in G, and ∞ otherwise. Motivated by applications to property reconstruction and access control hierarchies, we conce...

Given a complete graph [Formula: see text], with nonnegative edge costs, two subsets text] and partition of clustered Steiner tree is that spans all vertices in such can be cut into subtrees by removing edges each subtree text]. The cost defined to the sum costs its edges. A selected-internal for if are internal problem concerned determination minimum cost. In this paper, we present first known...

Intrinsic volumes and Minkowski tensors have been used to describe the geometry of real world objects. This paper presents an estimator that allows to approximate these quantities from digital images. It is based on a generalized Steiner formula for Minkowski tensors of sets of positive reach. When the resolution goes to infinity, the estimator converges to the true value if the underlying obje...

We introduce support (curvature) measures of an arbitrary closed set A in R and establish a local Steiner–type formula for the localized parallel volume of A. We derive some of the basic properties of these support measures and explore how they are related to the curvature measures available in the literature. Then we use the support measures in analysing contact distributions of stationary ran...

This paper establishes expectation and variance asymptotics for statistics of the Poisson-Voronoi approximation of general sets, as the underlying intensity of the Poisson point process tends to infinity. Statistics of interest include volume, surface area, Hausdorff measure, and the number of faces of lower-dimensional skeletons. We also consider the complexity of the so-called Voronoi zone an...

Richard Wilson conjectured in 1974 the following asymptotic formula for the number of n-vertex Steiner triple systems: STS(n) = ( (1 + o(1)) n e2 )n2 6 . Our main result is that STS(n) ≤ ( (1 + o(1)) n e2 )n2 6 . The proof is based on the entropy method. As a prelude to this proof we consider the number F (n) of 1factorizations of the complete graph on n vertices. Using the KahnLovász theorem i...

This paper contributes to the discrete differential geometry of triangle meshes, in combination with a study of discrete line congruences associated with such meshes. In particular we discuss when a congruence defined by linear interpolation of vertex normals deserves to be called a ‘normal’ congruence. Our main results are a discussion of various definitions of normality, a detailed study of t...

For a nonempty set S of vertices of a connected graph G, the Steiner distance d(S) of S is the minimum size among all connected subgraphs whose vertex set contains S. For an ordered set W = {Wl, W2,"', Wk} of vertices in a connected graph G and a vertex v of G, the Steiner representation s(vIW) of v with respect to W is the (2k I)-vector where d i1 ,i2, ... ,ij(V) is the Steiner distance d({V,W...

We present a general rectilinear Steiner tree problem in the plane and prove that it is solvable on the Hanan grid of the input points. This result is then used to show that several variants of the ordinary rectilinear Steiner tree problem are solvable on the Hanan grid, including | but not limited to | Steiner trees for rectilinear (or iso-thetic) polygons, obstacle-avoiding Steiner trees, gro...