Related papers: Space Complexity of Vertex Connectivity Oracles

A classic data structure problem is to preprocess a string T of length $n$ so that, given a query $q$, we can quickly find all substrings of T with Hamming distance at most $k$ from the query string. Variants of this problem have seen…

A (1 + eps)-approximate distance oracle for a graph is a data structure that supports approximate point-to-point shortest-path-distance queries. The most relevant measures for a distance-oracle construction are: space, query time, and…

Given a finite, simple graph $G$, the $k$-component order edge connectivity of $G$ is the minimum number of edges whose removal results in a subgraph for which every component has order at most $k-1$. In general, determining the…

The vertex connectivity of an $m$-edge $n$-vertex undirected graph is the smallest number of vertices whose removal disconnects the graph, or leaves only a singleton vertex. In this paper, we give a reduction from the vertex connectivity…

In the Survivable Network Design problem (SNDP), we are given an undirected graph $G(V,E)$ with costs on edges, along with a connectivity requirement $r(u,v)$ for each pair $u,v$ of vertices. The goal is to find a minimum-cost subset $E^*$…

Consider an undirected weighted graph G=(V,E) with |V|=n and |E|=m, where each vertex v is assigned a label from a set L of \ell labels. We show how to construct a compact distance oracle that can answer queries of the form: "what is the…

We present the first compact distance oracle that tolerates multiple failures and maintains exact distances. Given an undirected weighted graph $G = (V, E)$ and an arbitrarily large constant $d$, we construct an oracle that given vertices…

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,…

Let $\kappa(s,t)$ denote the maximum number of internally disjoint $st$-paths in an undirected graph $G$. We consider designing a compact data structure that answers $k$-bounded node connectivity queries: given $s,t \in V$ return…

Let $G$ be a nontrivial connected graph of order $n$ and let $k$ be an integer with $2\leq k\leq n$. For a set $S$ of $k$ vertices of $G$, let $\kappa (S)$ denote the maximum number $\ell$ of edge-disjoint trees $T_1,T_2,...,T_\ell$ in $G$…

The connectivity of a graph is an important parameter to evaluate its reliability. $k$-restricted connectivity (resp. $R^h$-restricted connectivity) of a graph $G$ is the minimum cardinality of a set $S$ of vertices in $G$, if exists, whose…

Connectivity (or equivalently, unweighted maximum flow) is an important measure in graph theory and combinatorial optimization. Given a graph $G$ with vertices $s$ and $t$, the connectivity $\lambda(s,t)$ from $s$ to $t$ is defined to be…

We study the distributed message-passing model in which a communication network is represented by a graph G=(V,E). Usually, the measure of complexity that is considered in this model is the worst-case complexity, which is the largest number…

We provide a deterministic algorithm for computing the $5$-edge-connected components of an undirected multigraph in linear time. There were probably good indications that this computation can be performed in linear time, but no such…

Our input is an undirected weighted graph $G = (V,E)$ on $n$ vertices along with a source set $S\subseteq V$. The problem is to preprocess $G$ and build a compact data structure such that upon query $Qu(s,v,f)$ where $(s,v) \in S\times V$…

We give an algorithm to compute all the local peaks in the $k$-level of an arrangement of $n$ lines in $O(n \log n) + \tilde{O}((kn)^{2/3})$ time. We can also find $\tau$ largest peaks in $O(n \log ^2 n) + \tilde{O}((\tau n)^{2/3})$ time.…

We consider distance queries in vertex-labeled planar graphs. For any fixed $0 < \epsilon \leq 1/2$ we show how to preprocess a directed planar graph with vertex labels and arc lengths into a data structure that answers queries of the…

We devise a data structure that can answer shortest path queries for two query points in a polygonal domain $P$ on $n$ vertices. For any $\varepsilon > 0$, the space complexity of the data structure is $O(n^{10+\varepsilon })$ and queries…

The \textsc{$l$-Exact Component Order Connectivity} problem asks whether, given an input graph $G$ and an integer $k$, there exists a vertex subset $S\subseteq V(G)$ of size at most $k$ such that every connected component in $G - S$ has…

An approximate distance oracle is a succinct data structure that provides fast answers to distance queries between any two nodes. In this paper we consider approximate distance oracles for general undirected graphs with non-negative edge…