Related papers: Approximate Distance Oracles for Planar Graphs wit…

In the minimum planarization problem, given some $n$-vertex graph, the goal is to find a set of vertices of minimum cardinality whose removal leaves a planar graph. This is a fundamental problem in topological graph theory. We present a…

Given an $n$-vertex undirected graph $G=(V,E,w)$, and a parameter $k\geq1$, a path-reporting distance oracle (or PRDO) is a data structure of size $S(n,k)$, that given a query $(u,v)\in V^2$, returns an $f(k)$-approximate shortest $u-v$…

We describe a new data structure for dynamic nearest neighbor queries in the plane with respect to a general family of distance functions. These include $L_p$-norms and additively weighted Euclidean distances. Our data structure supports…

There are several known data structures that answer distance queries between two arbitrary vertices in a planar graph. The tradeoff is among preprocessing time, storage space and query time. In this paper we present three data structures…

In this paper we present an efficient reachability oracle under single-edge or single-vertex failures for planar directed graphs. Specifically, we show that a planar digraph $G$ can be preprocessed in $O(n\log^2{n}/\log\log{n})$ time,…

An \emph{$\alpha$-approximate vertex fault-tolerant distance sensitivity oracle} (\emph{$\alpha$-VSDO}) for a weighted input graph $G=(V, E, w)$ and a source vertex $s \in V$ is the data structure answering an $\alpha$-approximate distance…

We show that the eccentricities, diameter, radius, and Wiener index of an undirected $n$-vertex graph with nonnegative edge lengths can be computed in time $O(n\cdot \binom{k+\lceil\log n\rceil}{k} \cdot 2^k k^2 \log n)$, where $k$ is the…

We consider the problem of preprocessing two strings $S$ and $T$, of lengths $m$ and $n$, respectively, in order to be able to efficiently answer the following queries: Given positions $i,j$ in $S$ and positions $a,b$ in $T$, return the…

We give an $O(n \log \log n)$ time algorithm for computing the minimum cut (or equivalently, the shortest cycle) of a weighted directed planar graph. This improves the previous fastest $O(n\log^3 n)$ solution. Interestingly, while in…

A distance oracle (DO) with stretch $(\alpha, \beta)$ for a graph $G$ is a data structure that, when queried with vertices $s$ and $t$, returns a value $\widehat{d}(s,t)$ such that $d(s,t) \le \widehat{d}(s,t) \le \alpha \cdot d(s,t) +…

In machine learning and particularly in topological data analysis, $\epsilon$-graphs are important tools but are generally hard to compute as the distance calculation between n points takes time O(n^2) classically. Recently, quantum…

It is unlikely that the discrete Fr\'echet distance between two curves of length $n$ can be computed in strictly subquadratic time. We thus consider the setting where one of the curves, $P$, is known in advance. In particular, we wish to…

$\renewcommand{\Re}{\mathbb{R}}$Given a set $P$ of $n$ points in $\Re^d$, and a parameter $\varepsilon \in (0,1)$, we present a new construction of a directed graph $G$, of size $O(n/\varepsilon^d)$, such that $(1+\varepsilon)$-ANN queries…

Consider a pair of plane straight-line graphs, whose edges are colored red and blue, respectively, and let n be the total complexity of both graphs. We present a O(n log n)-time O(n)-space technique to preprocess such pair of graphs, that…

The "short cycle removal" technique was recently introduced by Abboud, Bringmann, Khoury and Zamir (STOC '22) to prove fine-grained hardness of approximation. Its main technical result is that listing all triangles in an $n^{1/2}$-regular…

Given access to the vertex set $V$ of a connected graph $G=(V,E)$ and an oracle that given two vertices $u,v\in V$, returns the shortest path distance between $u$ and $v$, how many queries are needed to reconstruct $E$? Firstly, we show…

It is known that a better than $2$-approximation algorithm for the girth in dense directed unweighted graphs needs $n^{3-o(1)}$ time unless one uses fast matrix multiplication. Meanwhile, the best known approximation factor for a…

We study approximation algorithms for the following geometric version of the maximum coverage problem: Let $\mathcal{P}$ be a set of $n$ weighted points in the plane. Let $D$ represent a planar object, such as a rectangle, or a disk. We…

We study the Euclidean minimum weight perfect matching problem for $n$ points in the plane. It is known that any deterministic approximation algorithm whose approximation ratio depends only on $n$ requires at least $\Omega(n \log n)$ time.…

We develop a new algorithmic technique that allows to transfer some constant time approximation algorithms for general graphs into random order streaming algorithms. We illustrate our technique by proving that in random order streams with…