This short note describes a development of the traditional Pythagorean tree fractal to produce interesting three-dimensional structures based on cubes rather than squares. Adding a 90˚rotation at each bifurcation encourages three-dimensional growth, creating increasingly artistic shapes as the branching angle decreases. 1. Pythagorean box trees The Pythagorean tree is a plane fractal constructe...

We present worst-case lower bounds on the minimum size of a binary space partition (BSP) tree as a function of its height, for a set S of n axis-parallel line segments in the plane. We assume that the BSP uses only axis-parallel cutting lines. These lower bounds imply that, in the worst case, a BSP tree of height O(log n) must have size Ω(n logn) and a BSP tree of size O(n) must have height Ω(n...

In this paper, we study a Steiner tree related problem called “Terminal Steiner Tree with Bounded Edge Length”: given a set of terminal points P in a plane, one is asked to find a Steiner tree T such that any point in P is a leaf in T and the length of each edge in T is no more than a constant b. The objective of the problem is to minimize the number of Steiner points in T . This problem is mot...

We introduce a new BSP tree construction method for set of segments in the plane. Our algorithm is able to create BSP tree of linear size in the time O(n log n) for set of segments with low directional density (i.e. it holds for arbitrary segment si from such set, that a line created as extension of this segment doesn’t intersect too many other segments from the set in a near neighbourhood of s...

Several shape similarity measures, based on shape skeletons, are designed in the context of graph kernels. State-of-the-art kernels act on bags of walks, paths or trails which decompose the skeleton graph, and take into account structural noise through edition mechanisms. However, these approaches fail to capture the complexity of junctions inside skeleton graphs due to the linearity of the pat...

Abstract. Let T be a plane rooted tree with n nodes which is regarded as family tree of a Galton-Watson branching process conditioned on the total progeny. The profile of the tree may be described by the number of nodes or the number of leaves in layer t √ n, respectively. It is shown that these two processes converge weakly to Brownian excursion local time. This is done via characteristic func...

Let P and S be two disjoint sets of n and m points in the plane, respectively. We consider the problem of computing a Steiner tree whose Steiner vertices belong to S, in which each point of P is a leaf, and whose longest edge length is minimum. We present an algorithm that computes such a tree in O((n +m) logm) time, improving the previously best result by a logarithmic factor. We also prove a ...

Let G be a set of disjoint bi-chromatic straight line segments and H be a set of red and blue points in the plane, no three points are collinear. We give tight upper bounds on the maximum degree of a node in the color conforming minimum weight spanning tree (MST) formed by G and H. We also consider bounds on the total length of the edges of 1) the planar MST and the unrestricted MST, 2) the gre...

The Euclidean Steiner tree problem is solved by finding the tree with minimal Euclidean length spanning a set of fixed vertices in the plane, while allowing for the addition of auxiliary vertices (Steiner vertices). Steiner trees are widely used to design real-world structures like highways and oil pipelines. Unfortunately, the Euclidean Steiner Tree Problem has shown to be NP-Hard, meaning the...

In the minimum-cost k-hop spanning tree (k-hop MST) problem, we are given a set S of n points in a metric space, a positive small integer k and a root point r ∈ S. We are interested in computing a rooted spanning tree of minimum cost such that the longest root-leaf path in the tree has at most k edges. We present a polynomial-time approximation scheme for the plane. Our algorithm is based on Ar...