Rotational tree structures on binary trees and triangulations

A rotation in a binary tree is a simple and local restructuring technique commonly used in computer science. We propose in this paper three restrictions on the general rotation operation. We study the case when only leftmost rotations are permitted, which corresponds to a natural flipping on polygon triangulations. The resulting combinatorial structure is a tree structure with the root as the greatest element. We exhibit an efficient algorithm for computing the join of two trees and the minimum number of leftmost rotations necessary to transform one tree into the other

Random recursive triangulations of the disk via fragmentation theory

We introduce and study an infinite random triangulation of the unit disk that arises as the limit of several recursive models. This triangulation is generated by throwing chords uniformly at random in the unit disk and keeping only those chords that do not intersect the previous ones. After throwing infinitely many chords and taking the closure of the resulting set, one gets a random compact subset of the unit disk whose complement is a countable union of triangles. We show that this limiting random set has Hausdorff dimension $\beta^*+1$, where $\beta^*=(\sqrt{17}-3)/2$, and that it can be described as the geodesic lamination coded by a random continuous function which is H\"{o}lder continuous with exponent $\beta^*-\varepsilon$, for every $\varepsilon>0$. We also discuss recursive constructions of triangulations of the $n$-gon that give rise to the same continuous limit when $n$ tends to infinity.Comment: Published in at http://dx.doi.org/10.1214/10-AOP608 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Localization in Unstructured Environments: Towards Autonomous Robots in Forests with Delaunay Triangulation

Autonomous harvesting and transportation is a long-term goal of the forest industry. One of the main challenges is the accurate localization of both vehicles and trees in a forest. Forests are unstructured environments where it is difficult to find a group of significant landmarks for current fast feature-based place recognition algorithms. This paper proposes a novel approach where local observations are matched to a general tree map using the Delaunay triangularization as the representation format. Instead of point cloud based matching methods, we utilize a topology-based method. First, tree trunk positions are registered at a prior run done by a forest harvester. Second, the resulting map is Delaunay triangularized. Third, a local submap of the autonomous robot is registered, triangularized and matched using triangular similarity maximization to estimate the position of the robot. We test our method on a dataset accumulated from a forestry site at Lieksa, Finland. A total length of 2100\,m of harvester path was recorded by an industrial harvester with a 3D laser scanner and a geolocation unit fixed to the frame. Our experiments show a 12\,cm s.t.d. in the location accuracy and with real-time data processing for speeds not exceeding 0.5\,m/s. The accuracy and speed limit is realistic during forest operations

Bounds on the maximum multiplicity of some common geometric graphs

We obtain new lower and upper bounds for the maximum multiplicity of some weighted and, respectively, non-weighted common geometric graphs drawn on n points in the plane in general position (with no three points collinear): perfect matchings, spanning trees, spanning cycles (tours), and triangulations. (i) We present a new lower bound construction for the maximum number of triangulations a set of n points in general position can have. In particular, we show that a generalized double chain formed by two almost convex chains admits {\Omega}(8.65^n) different triangulations. This improves the bound {\Omega}(8.48^n) achieved by the double zig-zag chain configuration studied by Aichholzer et al. (ii) We present a new lower bound of {\Omega}(12.00^n) for the number of non-crossing spanning trees of the double chain composed of two convex chains. The previous bound, {\Omega}(10.42^n), stood unchanged for more than 10 years. (iii) Using a recent upper bound of 30^n for the number of triangulations, due to Sharir and Sheffer, we show that n points in the plane in general position admit at most O(68.62^n) non-crossing spanning cycles. (iv) We derive lower bounds for the number of maximum and minimum weighted geometric graphs (matchings, spanning trees, and tours). We show that the number of shortest non-crossing tours can be exponential in n. Likewise, we show that both the number of longest non-crossing tours and the number of longest non-crossing perfect matchings can be exponential in n. Moreover, we show that there are sets of n points in convex position with an exponential number of longest non-crossing spanning trees. For points in convex position we obtain tight bounds for the number of longest and shortest tours. We give a combinatorial characterization of the longest tours, which leads to an O(nlog n) time algorithm for computing them

Counting Carambolas

We give upper and lower bounds on the maximum and minimum number of geometric configurations of various kinds present (as subgraphs) in a triangulation of $n$ points in the plane. Configurations of interest include \emph{convex polygons}, \emph{star-shaped polygons} and \emph{monotone paths}. We also consider related problems for \emph{directed} planar straight-line graphs.Comment: update reflects journal version, to appear in Graphs and Combinatorics; 18 pages, 13 figure

The dual tree of a recursive triangulation of the disk

In the recursive lamination of the disk, one tries to add chords one after another at random; a chord is kept and inserted if it does not intersect any of the previously inserted ones. Curien and Le Gall [Ann. Probab. 39 (2011) 2224-2270] have proved that the set of chords converges to a limit triangulation of the disk encoded by a continuous process $\mathscr{M}$. Based on a new approach resembling ideas from the so-called contraction method in function spaces, we prove that, when properly rescaled, the planar dual of the discrete lamination converges almost surely in the Gromov-Hausdorff sense to a limit real tree $\mathscr{T}$, which is encoded by $\mathscr{M}$. This confirms a conjecture of Curien and Le Gall.Comment: Published in at http://dx.doi.org/10.1214/13-AOP894 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Finding Paths in the Rotation Graph of Binary Trees

A binary tree coding scheme is a bijection mapping a set of binary trees to a set of integer tuples called codewords. One problem considered in the literature is that of listing the codewords for n-node binary trees, such that successive codewords represent trees differing by a single rotation, a standard operation for rebalancing binary search trees. Then, the codeword sequence corresponds to an Hamiltonian path in the rotation graph Rn of binary trees, where each node is labelled with an n-node binary tree, and an edge connects two nodes when their trees differ by a single rotation. A related problem is finding a shortest path between two nodes in Rn, which reduces to the problem of transforming one binary tree into another using a minimum number of rotations. Yet a third problem is determining properties of the rotation graph. Our work addresses these three problems. A correspondence between n-node binary trees and triangulations of (n+2)-gons allows labelling nodes of Rn, with triangulations, where adjacent triangulations differ by a single diagonal flip. It has been proven, using properties of triangulations, that Rn is Hamiltonian, and that its diameter is bounded above by 2n-6 for n ≥ 11. In Chapter Three we use triangulations to show that the radius of Rn, is n-1; to characterize the n+2 nodes in the center; to show that Rn is the union of n+2 copies of Rn-1; and to prove that Rn is (n-1)-connected. We also introduce the skeleton graph RSn of Rn, and give additional properties of both graphs. In Chapter Four, we give an algorithm, OzLex, which, for each of many different coding schemes, generates 2n-1 different sequences of codewords for n-node binary trees. We also show that, for every n ≥ 4, all such sequences combined represent 2n Hamiltonian paths in Rn. In Appendix Two, we modify OzLex to create TransOx, an algorithm which generates (n+2)2n sequences of codewords from a single coding scheme, and prove that, for n ≥ 5, the sequences represent (n+2)2n-1 Hamiltonian paths. The distance between extreme nodes in Rn is the diameter of the graph. In Chapter Five, we give properties of extreme nodes in terms of their corresponding triangulations; Appendix One contains additional related information. We present two heuristics, based on flipping diagonals, that find a path between two nodes in Rn: Findpath-1, in O(n log n) time; and FindPath-2, in 0(n2 log n) time. Each computes paths with less than twice the minimum length. We also identify a class of triangulation pairs where Findpath-2 significantly outperforms FindPath-1

Spatial arrangements in architecture and mechanical engineering: some aspects of their representation and construction

Spatial arrangements in architecture and mechanical engineering are represented by incidence structures and classified according to properties of these incidence structures. The relationships between classes are given by ornamentation operations and the construction of elements in fundamental classes by substructure replacement operations. Thus representations of the spatial arrangements for possible designs are generated. Planar maps represent spatial arrangements in architecutral plans. The edges correspond to walls and vertices to incidence between walls. Plans represented by 3-vertex connected maps are ornamented by rooting and extension operations. Further ornamentation specifies access between regions. Plans with all regions adjacent to the exterior correspond to outerplane maps. Trivalent maps represent an important class of plans. Fundamental plans with r internal regions and s regions adjacent to the exterior are represented by [r,s] triangulations. Ornamentations of simple [r,s] triangulations are specified which represent plans with rectangular regions. Plans with walls aligned along two directions are represented by rectangular shapes whose maximal lines correspond to contiguous aligned walls. Rules of construction for various classes are given and the incidence structures of maximal lines and regions are characterized. Spatial arrangements in machines are represented by systems whose blocks correspond to links and vertices to joints. The dual systems are also used. Coplanar kinematic chains with revolute pairs are classified according to mobility and connectedness. Two fundamental classes are considered. First, the chains with binary joints, represented by simple graphs and constructed by two new methods: (i) suspended chain and cycle addition and (ii) subgraph replacement. Second, the chains with binary links which are constructed by subgraph replacement