Related papers: Light, Reliable Spanners
Given a set $S$ of $n$ points in the plane and a parameter $\varepsilon>0$, a Euclidean $(1+\varepsilon)$-spanner is a geometric graph $G=(S,E)$ that contains, for all $p,q\in S$, a $pq$-path of weight at most $(1+\varepsilon)\|pq\|$. We…
Given a 2-edge connected, unweighted, and undirected graph $G$ with $n$ vertices and $m$ edges, a $\sigma$-tree spanner is a spanning tree $T$ of $G$ in which the ratio between the distance in $T$ of any pair of vertices and the…
A rooted tree is balanced if the degree of a vertex depends only on its distance to the root. In this paper we determine the sharp threshold for the appearance of a large family of balanced spanning trees in the random geometric graph…
A Euclidean noncrossing Steiner $(1+\epsilon)$-spanner for a point set $P\subset\mathbb{R}^2$ is a planar straight-line graph that, for any two points $a, b \in P$, contains a path whose length is at most $1+\epsilon$ times the Euclidean…
A general formulation is presented for continuum scaling limits of stochastic spanning trees. A spanning tree is expressed in this limit through a consistent collection of subtrees, which includes a tree for every finite set of endpoints in…
We study the problem of maintaining a lightweight bounded-degree $(1+\varepsilon)$-spanner of a dynamic point set in a $d$-dimensional Euclidean space, where $\varepsilon>0$ and $d$ are arbitrary constants. In our fully-dynamic setting,…
A geometric $t$-spanner for a set $S$ of $n$ point sites is an edge-weighted graph for which the (weighted) distance between any two sites $p,q \in S$ is at most $t$ times the original distance between $p$ and~$q$. We study geometric…
Given a point set $P$ in the Euclidean space, a geometric $t$-spanner $G$ is a graph on $P$ such that for every pair of points, the shortest path in $G$ between those points is at most a factor $t$ longer than the Euclidean distance between…
Given an undirected possibly weighted $n$-vertex graph $G=(V,E)$ and a set $\mathcal{P}\subseteq V^2$ of pairs, a subgraph $S=(V,E')$ is called a ${\cal P}$-pairwise $\alpha$-spanner of $G$, if for every pair $(u,v)\in\mathcal{P}$ we have…
In this paper we address the problem of computing a sparse subgraph of a weighted directed graph such that the exact distances from a designated source vertex to all other vertices are preserved under bounded weight increment. Finding a…
Solomon and Elkin constructed a shortcutting scheme for weighted trees which results in a 1-spanner for the tree metric induced by the input tree. The spanner has logarithmic lightness, logarithmic diameter, a linear number of edges and…
We consider the probability that a spanning tree chosen uniformly at random from a graph can be partitioned into a fixed number $k$ of trees of equal size by removing $k-1$ edges. In that case, the spanning tree is called {\em splittable}.…
Let $\Omega$ be an open half-space or slab in $\mathbb{R}^{n+1}$ endowed with a perturbation of the Gaussian measure of the form $f(p):=\exp(\omega(p)-c|p|^2)$, where $c>0$ and $\omega$ is a smooth concave function depending only on the…
Designing well-connected graphs is a fundamental problem that frequently arises in various contexts across science and engineering. The weighted number of spanning trees, as a connectivity measure, emerges in numerous problems and plays a…
In STOC'95 Arya et al. (1995) conjectured that for any constant dimensional $n$-point Euclidean space, a $(1+\eps)$-spanner with constant degree, hop-diameter $O(\log n)$ and weight $O(\log n) \cdot \omega(MST)$ can be built in $O(n \log…
Stability and dependence are model-theoretic notions that have recently proved highly effective in the study of structural and algorithmic properties of hereditary graph classes, and are considered key notions for generalizing to hereditary…
We study the type of solutions to which stochastic gradient descent converges when used to train a single hidden-layer multivariate ReLU network with the quadratic loss. Our results are based on a dynamical stability analysis. In the…
We study the problem of how well a tree metric is able to preserve the sum of pairwise distances of an arbitrary metric. This problem is closely related to low-stretch metric embeddings and is interesting by its own flavor from the line of…
A tree $t$-spanner of a graph $G$ is a spanning tree of $G$ such that the distance between pairs of vertices in the tree is at most $t$ times their distance in $G$. Deciding tree $t$-spanner admissible graphs has been proved to be tractable…
Graph sparsification has been studied extensively over the past two decades, culminating in spectral sparsifiers of optimal size (up to constant factors). Spectral hypergraph sparsification is a natural analogue of this problem, for which…