Tree Embedding Research Articles

Markov convexity and nonembeddability of the Heisenberg group

We compute the Markov convexity invariant of the continuous infinite dimensional Heisenberg group $\mathbb{H}_\infty$ to show that it is Markov 4-convex and cannot be Markov $p$-convex for any $p < 4$. As Markov convexity is a biLipschitz invariant and Hilbert space is Markov 2-convex, this gives a different proof of the classical theorem of Pansu and Semmes that the Heisenberg group does not admit a biLipschitz embedding into any Euclidean space. The Markov convexity lower bound will follow from exhibiting an explicit embedding of Laakso graphs $G_n$ into $\mathbb{H}_\infty$ that has distortion at most $C n^{1/4} \sqrt{\log n}$. We use this to show that if $X$ is a Markov $p$-convex metric space, then balls of the discrete Heisenberg group $\mathbb{H}(\mathbb{Z})$ of radius $n$ embed into $X$ with distortion at least some constant multiple of $$\frac{(\log n)^{\frac{1}{p}-\frac{1}{4}}}{\sqrt{\log \log n}}.$$ Finally, we show that Markov 4-convexity does not give the optimal distortion for embeddings of binary trees $B_m$ into $\mathbb{H}_\infty$ by showing that the distortion is on the order of $\sqrt{\log m}$.

The approximate Loebl-Komlós-Sós conjecture and embedding trees in sparse graphs

Loebl, Koml\'os and S\'os conjectured that every $n$-vertex graph $G$ with at least $n/2$ vertices of degree at least $k$ contains each tree $T$ of order $k+1$ as a subgraph. We give a sketch of a proof of the approximate version of this conjecture for large values of $k$. For our proof, we use a structural decomposition which can be seen as an analogue of Szemer\'edi's regularity lemma for possibly very sparse graphs. With this tool, each graph can be decomposed into four parts: a set of vertices of huge degree, regular pairs (in the sense of the regularity lemma), and two other objects each exhibiting certain expansion properties. We then exploit the properties of each of the parts of $G$ to embed a given tree $T$. The purpose of this note is to highlight the key steps of our proof. Details can be found in [arXiv:1211.3050].

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

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 $\frac{n}{3}$ 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.

Embedding of Binomial Trees in Locally Twisted Cubes with Link Faults

As a newly introduced interconnection network for parallel computing, the locally twisted cube possesses many desirable properties. In this paper, binomial tree embeddings in locally twisted cubes are studied. We present two major results in this paper: (1) For any integern≥ 2, ann-dimensional binomial treeBncan be embedded inLTQnwith dilation 1 by randomly choosing any vertex inLTQnas the root. (2) For any integern≥ 2, ann-dimensional binomial treeBncan be embedded inLTQnwith up ton− 1 faulty links inlog(n− 1) steps where dilation = 1. The results are optimal in the sense that the dilations of all embeddings are 1.

Yet another short proof of Bourgain’s distortion estimate for embedding of trees into uniformly convex Banach spaces

We use a self-improvement argument to give a very short and elementary proof of the result of Bourgain saying that infinite regular trees do not admit bi-Lipschitz embeddings into uniformly convex Banach spaces.

Random Walks on Quasirandom Graphs

Let $G$ be a quasirandom graph on $n$ vertices, and let $W$ be a random walk on $G$ of length $\alpha n^2$. Must the set of edges traversed by $W$ form a quasirandom graph? This question was asked by Böttcher, Hladký, Piguet and Taraz. Our aim in this paper is to give a positive answer to this question. We also prove a similar result for random embeddings of trees.

On the Tree Packing Conjecture

The Gyárfás tree packing conjecture states that any set of $n-1$ trees $T_{1},T_{2},\dots, T_{n-1}$ such that $T_i$ has $n-i+1$ vertices packs into $K_n$ (for $n$ large enough). We show that $t=\frac{1}{10}n^{1/4}$ trees $T_1,T_2,\dots, T_t$ such that $T_i$ has $n-i+1$ vertices packs into $K_{n+1}$ (for $n$ large enough). We also prove that any set of $t=\frac{1}{10}n^{1/4}$ trees $T_1,T_2,\dots, T_t$ such that no tree is a star and $T_i$ has $n-i+1$ vertices packs into $K_{n}$ (for $n$ large enough). Finally, we prove that $t=\frac{1}{4}n^{1/3}$ trees $T_1,T_2,\dots, T_t$ such that $T_i$ has $n-i+1$ vertices packs into $K_n$ as long as each tree has maximum degree at least $2n^{2/3}$ (for $n$ large enough). One of the main tools used in the paper is the famous spanning tree embedding theorem of Komlós, Sárközy, and Szemerédi [Combin. Probab. Comput., 10 (2001), pp. 397--416].

On the area requirements of Euclidean minimum spanning trees

In their seminal paper on Euclidean minimum spanning trees, Monma and Suri (1992) proved that any tree of maximum degree 5 admits a planar embedding as a Euclidean minimum spanning tree. Their algorithm constructs embeddings with exponential area; however, the authors conjectured that there exist n-vertex trees of maximum degree 5 that require cn×cn area to be embedded as Euclidean minimum spanning trees, for some constant c>1. In this paper, we prove the first exponential lower bound on the area requirements for embedding trees as Euclidean minimum spanning trees.

Tree representation of digital picture embeddings

It is often the case that the same object is imaged in different ways, resulting in digital pictures of (some parts of) it at different resolutions. This leads to the combinatorial problem of “embedding” one of these pictures into the other in a way that corresponds to physical truth. In this paper we present a mathematical formulation of this intuitive concept of embedding. We also show, using a tree representation of digital pictures, how picture embedding relates to tree embedding, which has been a subject of much study in combinatorial computer science (mostly for reasons other than application to digital pictures).

Fast embedding of spanning trees in biased Maker–Breaker games

Given a tree T=(V,E) on n vertices, we consider the (1:q) Maker–Breaker tree embedding game Tn. The board of this game is the edge set of the complete graph on n vertices. Maker wins Tn if and only if she is able to claim all edges of a copy of T. We prove that there exist real numbers α,ε>0 such that, for sufficiently large n and for every tree T on n vertices with maximum degree at most nε, Maker has a winning strategy for the (1:q) game Tn, for every q≤nα. Moreover, we prove that Maker can win this game within n+o(n) moves which is clearly asymptotically optimal.

Optimal distortion embedding of complete binary trees into lines

We study the problem of embedding an unweighted complete binary tree into a line with low distortion. Very recently, Mathieu and Papamanthou (2008) [9] showed that the inorder traversal of the complete binary tree of height h gives a line embedding of distortion O(2h), and conjectured that the current lower bound of Ω(2hh) increases to Ω(2h), i.e., the upper bound of O(2h) is best possible. In this paper, we disprove their conjecture by providing a line embedding of the complete binary tree of height h with optimal distortion Θ(2hh).

Efficient distributed approximation algorithms via probabilistic tree embeddings

We present a uniform approach to design efficient distributed approximation algorithms for various fundamental network optimization problems. Our approach is randomized and based on a probabilistic tree embedding due to Fakcharoenphol et al. (J Comput Syst Sci 69(3):485–497, 2004) (FRT embedding). We show how to efficiently compute an (implicit) FRT embedding in a decentralized manner and how to use the embedding to obtain efficient expected O(log n)-approximate distributed algorithms for various problems, in particular the generalized Steiner forest problem (including the minimum Steiner tree problem), the minimum routing cost spanning tree problem, and the k-source shortest paths problem. The distributed construction of the FRT embedding is based on the computation of least elements (LE) lists, a distributed data structure that is of independent interest. Assuming a global order on the nodes of a network, the LE-list of a node stores the smallest node (w.r.t. the given order) within every distance d (cf. Cohen in J Comput Syst Sci 55(3):441–453, 1997, Cohen and Kaplan in J Comput Syst Sci 73(3):265–288, 2007). Assuming a random order on the nodes, we give a distributed algorithm for computing LE-lists on a weighted graph with time complexity O(S log n), where S is a graph parameter called the shortest path diameter which can be considered the weighted counterpart of the diameter D of the graph. For unweighted graphs, our LE-lists computation has asymptotically optimal time complexity of O(D). As a byproduct, we get an improved synchronous leader election algorithm for general networks that is both time-optimal and almost message-optimal with high probability.

Wirelength of hypercubes into certain trees

A lot of research has been devoted to finding efficient embedding of trees into hypercubes. On the other hand, in this paper, we consider the problem of embedding hypercubes into k-rooted complete binary trees, k-rooted sibling trees, binomial trees and certain classes of caterpillars to minimize the wirelength.

Fast embedding of spanning trees in biased Maker-Breaker games

Abstract Given a tree T = ( V , E ) on n vertices, we consider the ( 1 : q ) Maker-Breaker tree embedding game T n . The board of this game is the edge set of the complete graph on n vertices. Maker wins T n if and only if he is able to claim all edges of a copy of T. We prove that there exist real numbers α , e > 0 such that, for sufficiently large n and for every tree T on n vertices with maximum degree at most n e , Maker has a winning strategy for the ( 1 : q ) game T n , for every q ⩽ n α . Moreover, we prove that Maker can win this game within n + o ( n ) moves which is clearly asymptotically optimal.

Polynomial area bounds for MST embeddings of trees

In their seminal paper on geometric minimum spanning trees, Monma and Suri (1992) [31] showed how to embed any tree of maximum degree 5 as a minimum spanning tree in the Euclidean plane. The embeddings provided by their algorithm require area O ( 2 n 2 ) × O ( 2 n 2 ) and the authors conjectured that an improvement below c n × c n is not possible, for some constant c > 0 . In this paper, we show how to construct MST embeddings of arbitrary trees of maximum degree 3 and 4 within polynomial area.

Combinatorial theorems about embedding trees on the real line

We consider the combinatorial problem of embedding the metric defined by an unweighted graph into the real line, so as to minimize the distortion of the embedding. This problem is inspired by connections to Banach space theory and to computer science. After establishing a framework in which to study line embeddings, we focus on metrics defined by three specific families of trees: complete binary trees, fans, and combs. We construct asymptotically optimal (i.e., distortion-minimizing) line embeddings for these metrics and prove their optimality via suitable lower bound arguments. It appears that even such specialized metrics require nontrivial constructions and proofs of optimality require sophisticated combinatorial arguments. The combinatorial techniques from our work might prove useful in further algorithmic research on low distortion metric embeddings. © 2011 Wiley Periodicals, Inc. J Graph Theory 67: 153–168, 2011 © 2011 Wiley Periodicals, Inc.

Unordered tree matching and ordered tree matching: the evaluation of tree pattern queries

In this paper, we study the twig pattern matching in XML document databases. Two algorithms A1 and A2 are discussed according to two different definitions of tree embedding. By the first definition, only the ancestor-descendant relationship is considered. By the second one, we take not only the ancestor-descendant relationship, but also the order of siblings into account. Both A1 and A2 are based on a subtree reconstruction technique, by which a tree structure is reconstructed according to a given set of data streams. More importantly, by revealing an interesting property of tree encoding, we show that the subtree reconstruction can be easily extended to a strategy (i.e., A1) for checking subtree matching according to the first definition with any kind of path join or join-like operations being completely avoided. A2 needs more time and space since it deals with a more difficult problem, but without join operations involved, either. The computational complexities of both algorithms are analysed, showing that they have a better performance than any existing strategy for this problem.

Complexity dichotomy on degree-constrained VLSI layouts with unit-length edges

Deciding whether an arbitrary graph admits a VLSI layout with unit-length edges is NP-complete [Bhatt, S.N., and Cosmadakis, S.S., The complexity of minimizing wire lengths in VLSI layouts, Inform. Process. Lett. 25 (1987), 263–267], even when restricted to binary trees [Gregori, A., Unit-length embedding of binary trees on a square grid, Inform. Process. Lett. 31 (1989), 167–173]. However, for certain graphs, the problem is polynomial or even trivial. A natural step, outstanding thus far, was to provide a broader classification of graphs that make for polynomial or NP-complete instances. We provide such a classification based on the set of vertex degrees in the input graphs, yielding a comprehensive dichotomy on the complexity of the problem, with and without the restriction to trees.

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

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 point-set 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].

Some results on metric trees

Using isometric embedding of metric trees into Banach spaces, this paper will investigate barycenters, type and cotype, and various measures of compactness of metric trees. A&nbsp;metric tree ($T$, $d$) is a metric space such that between any two of its