Given a set S of n red and blue points in the plane, a planar bichromatic minimum spanning tree is the shortest possible spanning tree of S, such that every edge connects a red and a blue point, and no two edges intersect. Computing this tree is NP-hard in general. We present an O(n) time algorithm for the special case when all points are in convex position. For the general case, we present a f...

The network and Euclidean Steiner tree problems require a shortest tree spanning a given vertex subset within a network G = (V; E; d) and Euclidean plane, respectively. For these problems, we present a series of heuristics nding approximate Steiner tree with performance guarantee coming arbitrary close to 1+ln2 1:693 and 1+ln 2 p 3 1:1438, respectively. The best previously known corresponding v...

We study a bottleneck Steiner tree problem: given a set P = {p1,p2, . . . , pn} of n terminals in the Euclidean plane and a positive integer k, find a Steiner tree with at most k Steiner points such that the length of the longest edges in the tree is minimized. The problem has applications in the design of wireless communication networks. We give a ratio-1.866 approximation algorithm for the pr...

In this paper we are concerned with the analysis of special recursive algorithms for traversing the nodes of a planted plane tree (ordered tree ; planar tree) . Some by now classical results in this area are due to (de Bruijn, Knuth, and Rice, 1972 ; Knuth, 1973 ; Flajolet, 1981 ; Flajolet and Odlyzko, 1982 ; Flajolet, Raoult, and Vuillemin, 1979 ; Kemp, 1979) and others and are summarized in t...

We wish to decide whether a simply connected flexible polygonal structure can be realized in Euclidean space. Two models are considered: polygonal linkages (body-and-joint framework) and contact graphs of unit disks in the plane. (1) We show that it is strongly NP-hard to decide whether a given polygonal linkage is realizable in the plane when the bodies are convex polygons and their contact gr...

We introduce and study orderly spanning trees of plane graphs. This algorithmic tool generalizes canonical orderings, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an orderly pair for any connected planar graph G, consisting of an embedded planar graph H isomorphic to G, and an orderly spanning ...

A model of a rigid-plastic rate-independent polycrystalline aggregate wherein subaggregates are represented as the nodes of a binary tree is proposed. The lowest nodes of the binary tree represent grains. Higher binary tree nodes represent increasingly larger sub-aggregates of grains, culminating with the root of the tree, which represents the entire polycrystalline aggregate. Planar interfaces...

We introduce a flow-dependent version of the quadratic Steiner tree problem in the plane. An instance of the problem on a set of embedded sources and a sink asks for a directed tree T spanning these nodes and a bounded number of Steiner points, such that ∑

The Oriental Plane Tree (Platanus Orientalis), not to be confused with the better known London Plane Tree (Platanus Acerifolia), is one of Europe’s longest-lived trees. A native of SE Europe and Asia Minor, it is occasionally found in British parks and gardens, having been cultivated there since the sixteenth century. It can reach 100 feet in height and grows well in open ground, its branches, ...

In the book Boolean Function Complexity by Stasys Jukna [7], two lower bound techniques for Tree-like Cutting Plane proofs (henceforth, “Tree-CP proofs”) using Karchmer-Widgerson type communication games (henceforth, “KW games”) are presented: The first, applicable to Tree-CP proofs with bounded coefficients, translates Ω(t) deterministic lower bounds on KW games to 2 logn) lower bounds on Tree...