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A graph spanner is a fundamental graph structure that faithfully preserves the pairwise distances in the input graph up to a small multiplicative stretch. The common objective in the computation of spanners is to achieve the best-known…
A natural requirement of many distributed structures is fault-tolerance: after some failures, whatever remains from the structure should still be effective for whatever remains from the network. In this paper we examine spanners of general…
In fault-tolerant distance labeling we wish to assign short labels to the vertices of a graph $G$ such that from the labels of any three vertices $u,v,f$ we can infer the $u$-to-$v$ distance in the graph $G\setminus \{f\}$. We show that any…
In this paper we introduce a notion of spectral approximation for directed graphs. While there are many potential ways one might define approximation for directed graphs, most of them are too strong to allow sparse approximations in…
We provide an algorithm for computing a planar morph between any two planar straight-line drawings of any $n$-vertex plane graph in $O(n)$ morphing steps, thus improving upon the previously best known $O(n^2)$ upper bound. Further, we prove…
Graph Crossing Number is a fundamental problem with various applications. In this problem, the goal is to draw an input graph $G$ in the plane so as to minimize the number of crossings between the images of its edges. Despite extensive…
Given an n-vertex graph G, a drawing of G in the plane is a mapping of its vertices into points of the plane, and its edges into continuous curves, connecting the images of their endpoints. A crossing in such a drawing is a point where two…
In this work we provide a new technique to design fast approximation algorithms for graph problems where the points of the graph lie in a metric space. Specifically, we present a sampling approach for such metric graphs that, using a…
We initiate the study of approximation algorithms and computational barriers for constructing sparse $\alpha$-navigable graphs [IX23, DGM+24], a core primitive underlying recent advances in graph-based nearest neighbor search. Given an…
The Sparsest Cut is a fundamental optimization problem that has been extensively studied. For planar inputs the problem is in $P$ and can be solved in $\tilde{O}(n^3)$ time if all vertex weights are $1$. Despite a significant amount of…
In this paper, we consider the problem of clustering graph nodes and sparsifying graph edges over distributed graphs, when graph edges with possibly edge duplicates are observed at physically remote sites. Although edge duplicates across…
We introduce a new approach to spectral sparsification that approximates the quadratic form of the pseudoinverse of a graph Laplacian restricted to a subspace. We show that sparsifiers with a near-linear number of edges in the dimension of…
Effective Resistance (ER) is a fundamental tool in various graph learning tasks. In this paper, we address the problem of efficiently approximating ER on a graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ with $n$ vertices and $m$ edges.…
We (nearly) settle the time complexity for computing vertex fault-tolerant (VFT) spanners with optimal sparsity (up to polylogarithmic factors). VFT spanners are sparse subgraphs that preserve distance information, up to a small…
The task of finding an extension to a given partial drawing of a graph while adhering to constraints on the representation has been extensively studied in the literature, with well-known results providing efficient algorithms for…
We give faster algorithms for producing sparse approximations of the transition matrices of $k$-step random walks on undirected, weighted graphs. These transition matrices also form graphs, and arise as intermediate objects in a variety of…
A hypergraph spectral sparsifier of a hypergraph $G$ is a weighted subgraph $H$ that approximates the Laplacian of $G$ to a specified precision. Recent work has shown that similar to ordinary graphs, there exist $\widetilde{O}(n)$-size…
Given a large edge-capacitated network $G$ and a subset of $k$ vertices called terminals, an (exact) flow sparsifier is a small network $G'$ that preserves (exactly) all multicommodity flows that can be routed between the terminals. Flow…
Compression and sparsification algorithms are frequently applied in a preprocessing step before analyzing or optimizing large networks/graphs. In this paper we propose and study a new framework contracting edges of a graph (merging vertices…
A $t$-spanner of a graph $G$ is a subgraph $H$ in which all distances are preserved up to a multiplicative $t$ factor. A classical result of Alth\"ofer et al. is that for every integer $k$ and every graph $G$, there is a $(2k-1)$-spanner of…