Optimal Algorithms to Embed Trees in a Point Set (original) (raw)

Planar embedding of trees on point sets without the general position assumption

TURKISH JOURNAL OF MATHEMATICS, 2015

The problem of point-set embedding of a planar graph G on a point set P in the plane is defined as finding a straight-line planar drawing of G such that the nodes of G are mapped one to one on the points of P. Previous works in this area mostly assume that the points of P are in general position, i.e. P does not contain any three collinear points. However, in most of the real applications we cannot assume the general position assumption. In this paper, we show that deciding the point-set embeddability of trees without the general position assumption is NP-complete. Then we introduce an algorithm for point-set embedding of n-node binary trees with at most n 3 total bends on any point set. We also give some results when the problem is limited to degree-constrained trees and point sets having constant number of collinear points.

Planar straight-line point-set embedding of trees with partial embeddings

Information Processing Letters, 2010

Given a set P of n points in the plane, an n-node tree T , and a partial embedding E of T on P (i.e. a planar straight-line point-set embedding of some sub-trees of T on a sub-set of P ), we show that the problem of deciding whether there is a planar straight-line pointset embedding of T on P that includes E is NP-complete. This problem was posed as an open problem in E. Di Giacomo et al. (2009) [8].

Heuristic Algorithms for Geometric Embedding of Complete Binary Trees onto a Point-set

In the geometric graph embedding problem, a graph with n vertices and a set of n points in the plane are given, and the aim of embedding is to find a mapping between vertices of the graph to these points in such a way that minimizes the length of the embedded graph on the point set. Since the travelling salesman problem is a special case of the graph embedding problem, therefore, the problem is an NPhard problem. In this paper, we consider a particular case where the given graph is a binary tree. We present four heuristic approaches, then we compare the time complexity, and the resulted embedding length of these algorithms.

Orthogeodesic point-set embedding of trees

2012

Let S be a set of N grid points in the plane, and let G a graph with n vertices (n ≤ N). An orthogeodesic point-set embedding of G on S is a drawing of G such that each vertex is drawn as a point of S and each edge is an orthogonal chain with bends on grid points whose length is equal to the Manhattan distance. We study the following problem. Given a family of trees F what is the minimum value f (n) such that every n-vertex tree in F admits an orthogeodesic point-set embedding on every grid-point set of size f (n)? We provide polynomial upper bounds on f (n) for both planar and non-planar orthogeodesic point-set embeddings as well as for the case when edges are required to be L-shaped chains.

Point-set embeddings of trees with given partial drawings

2009

Given a graph G with n vertices and a set S of n points in the plane, a point-set embedding of G on S is a planar drawing such that each vertex of G is mapped to a distinct point of S. A geometric point-set embedding is a point-set embedding with no edge bends. This paper studies the following problem: The input is a set S of n points, a planar graph G with n vertices, and a geometric point-set embedding of a subgraph G ⊂ G on a subset of S. The desired output is a point-set embedding of G on S that includes the given partial drawing of G . We concentrate on trees and show how to compute the output in O (n 2 log n) time in a real-RAM model and with at most n − k edges with at most 1 + 2 k/2 bends, where k is the number of vertices of the given subdrawing. We also prove that there are instances of the problem which require at least k − 3 bends on n − k edges.

Low-Distortion Embeddings of Trees

Journal of Graph Algorithms and Applications, 2003

We prove that every tree T = (V, E) on n vertices with edges of unit length can be embedded in the plane with distortion O(√ n); that is, we construct a mapping f : V → R 2 such that ρ(u, v) ≤ f (u) − f (v) ≤ O(√ n) • ρ(u, v) for every u, v ∈ V , where ρ(u, v) denotes the length of the path from u to v in T. The embedding is described by a simple and easily computable formula. This is asymptotically optimal in the worst case. We also construct interesting optimal embeddings for a special class of trees (fans consisting of paths of the same length glued together at a common vertex).

Optimal one-page tree embeddings in linear time

Information Processing Letters, 2003

In the minimum linear arrangement problem one wishes to assign distinct integers to the vertices of a given graph so that the sum of the differences (in absolute value) across the edges of the graph is minimized. This problem is known to be NP-complete for the class of all graphs, but polynomial for trees-algorithms of time complexity O(n 2.2 ) and O(n 1.6 ) were given by Shiloach [SIAM J. Comput. 8 (1979) 15-32] and Chung [Comput. Math. Appl. 10 (1984) 43-60], respectively. We present a linear-time algorithm for finding the optimal embedding (arrangement) in a restricted but important class of embeddings called one-page embeddings. 1

Point-set embeddings of plane 3-trees

Computational Geometry, 2012

A straight-line drawing of a plane graph G is a planar drawing of G, where each vertex is drawn as a point and each edge is drawn as a straight line segment. Given a set S of n points in the Euclidean plane, a point-set embedding of a plane graph G with n vertices on S is a straight-line drawing of G, where each vertex of G is mapped to a distinct point of S. The problem of deciding if G admits a point-set embedding on S is NP-complete in general and even when G is 2-connected and 2-outerplanar. In this paper, we give an O(n 2 ) time algorithm to decide whether a plane 3-tree admits a point-set embedding on a given set of points or not, and find an embedding if it exists. We prove an Ω(n log n) lower bound on the time complexity for finding a point-set embedding of a plane 3-tree. We then consider a variant of the problem, where we are given a plane 3-tree G with n vertices and a set S of k > n points, and present a dynamic programming algorithm to find a point-set embedding of G on S if it exists. Furthermore, we show that the point-set embeddability problem for planar partial 3-trees is also NP-complete.

Embedding Plane 3-Trees in ℝ2 and ℝ3

Lecture Notes in Computer Science, 2012

A point-set embedding of a planar graph G with n vertices on a set P of n points in R d , d ≥ 1, is a straight-line drawing of G, where the vertices of G are mapped to distinct points of P . The problem of computing a point-set embedding of G on P is NP-complete in R 2 , even when G is 2-outerplanar and the points are in general position. On the other hand, if the points of P are in general position in R 3 , then any bijective mapping of the vertices of G to the points of P determines a point-set embedding of G on P . In this paper, we give an O(n 4/3+ǫ )expected time algorithm to decide whether a plane 3-tree with n vertices admits a point-set embedding on a given set of n points in general position in R 2 and compute such an embedding if it exists, for any fixed ǫ>0. We extend our algorithm to embed a subclass of 4-trees on a point set in R 3 in the form of nested tetrahedra. We also prove that given a plane 3-tree G with n vertices, a set P of n points in R 3 that are not necessarily in general position and a mapping of the three outer vertices of G to three different points of P , it is NP-complete to decide if G admits a point-set embedding on P respecting the given mapping.