Related papers: Dimension reduction for finite trees in L_1

It is known that every graph with n vertices embeds stochastically into trees with distortion $O(\log n)$. In this paper, we show that this upper bound is sharp for a large class of graphs. As this class of graphs contains diamond graphs,…

Cohen-Addad, Le, Pilipczuk, and Pilipczuk [CLPP23] recently constructed a stochastic embedding with expected $1+\varepsilon$ distortion of $n$-vertex planar graphs (with polynomial aspect ratio) into graphs of treewidth…

We study deterministic online embeddings of metrics spaces into normed spaces and into trees against an adaptive adversary. Main results include a polynomial lower bound on the (multiplicative) distortion of embedding into Euclidean spaces,…

A decision tree recursively splits a feature space $\mathbb{R}^{d}$ and then assigns class labels based on the resulting partition. Decision trees have been part of the basic machine-learning toolkit for decades. A large body of work treats…

Treedepth is a central parameter to algorithmic graph theory. The current state-of-the-art in computing and approximating treedepth consists of a $2^{O(k^2)} n$-time exact algorithm and a polynomial-time $O(\text{OPT} \log^{3/2}…

It is known that any $n$-point set in the $d$-dimensional Euclidean space $\mathbb{R}^d$, for $d = O(1)$, admits: 1) a $(1+\epsilon)$-spanner with maximum degree $\tilde{O}(\epsilon^{-d+1})$ and with lightness $\tilde{O}(\epsilon^{-d})$; 2)…

An infinite dimensional notion of asymptotic structure is considered. This notion is developed in terms of trees and branches on Banach spaces. Every countably infinite countably branching tree $\mathcal T$ of a certain type on a space X is…

The sparse Johnson-Lindenstrauss transform is one of the central techniques in dimensionality reduction. It supports embedding a set of $n$ points in $\mathbb{R}^d$ into $m=O(\varepsilon^{-2} \lg n)$ dimensions while preserving all pairwise…

Diversities are like metric spaces, except that every finite subset, instead of just every pair of points, is assigned a value. Just as there is a theory of minimal distortion embeddings of finite metric spaces into $L_1$, there is a…

The Subtree Isomorphism problem asks whether a given tree is contained in another given tree. The problem is of fundamental importance and has been studied since the 1960s. For some variants, e.g., ordered trees, near-linear time algorithms…

Let $T$ be an $n$-node tree of maximum degree 4, and let $P$ be a set of $n$ points in the plane with no two points on the same horizontal or vertical line. It is an open question whether $T$ always has a planar drawing on $P$ such that…

We study upward planar straight-line embeddings (UPSE) of directed trees on given point sets. The given point set $S$ has size at least the number of vertices in the tree. For the special case where the tree is a path $P$ we show that: (a)…

For a fixed $K\gg 1$ and $n\in\mathbb{N}$, $n\gg 1$, we study metric spaces which admit embeddings with distortion $\le K$ into each $n$-dimensional Banach space. Classical examples include spaces embeddable into $\log n$-dimensional…

We show that an infinite weighted tree admits a bi-Lipschitz embedding into Hilbert space if and only if it does not contain arbitrarily large complete binary trees with uniformly bounded distortion. We also introduce a new metric invariant…

For any finite point set in $D$-dimensional space equipped with the 1-norm, we present random linear embeddings to $k$-dimensional space, with a new metric, having the following properties. For any pair of points from the point set that are…

We divide the class of infinite computable trees into three types. For the first and second types, $0'$ computes a nontrivial self-embedding while for the third type $0''$ computes a nontrivial self-embedding. These results are optimal and…

Embeddings of graphs into distributions of trees that preserve distances in expectation are a cornerstone of many optimization algorithms. Unfortunately, online or dynamic algorithms which use these embeddings seem inherently randomized and…

We prove that every visual Gromov hyperbolic space X whose boundary at infinity has the finite capacity dimension n admits a quasi-isometric embedding into (n+1)-fold product of metric trees.

It is shown that every tree of size $n$ over a fixed set of $\sigma$ different ranked symbols can be decomposed (in linear time as well as in logspace) into $O\big(\frac{n}{\log_\sigma n}\big) = O\big(\frac{n \log \sigma}{\log n}\big)$ many…

We prove that every $n$-point metric space of negative type (and, in particular, every $n$-point subset of $L_1$) embeds into a Euclidean space with distortion $O(\sqrt{\log n} \cdot\log \log n)$, a result which is tight up to the iterated…