Planar Reachability Under Single Vertex or Edge Failures

  title={Planar Reachability Under Single Vertex or Edge Failures},
  author={Giuseppe F. Italiano and Adam Karczmarz and Nikos Parotsidis},
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 n/log logn) time, producing an O(n log n)-space data structure that can answer in O(log n) time whether u can reach v in G if the vertex x (the edge f) is removed from G, for any query vertices u, v and failed vertex x (failed edge f). To the best of our knowledge, this is the first data… Expand

Figures from this paper

Fault-Tolerant Distance Labeling for Planar Graphs
Any directed weighted planar graph (and in fact any graph in a graph family with O( √ n)-size separators, such as minor-free graphs) admits fault-tolerant distance labels of size O(n2/3), and these labels are extended in a way that allows to also count the number of shortest paths. Expand


An Optimal Dual Fault Tolerant Reachability Oracle
It is shown that it is possible to compute in polynomial time an O(n) size data structure that for any query vertex v, and any pair of failed vertices f_1, f_2, answers in O(1) time whether or not there exists a path from s to v in G, and a labeling scheme with O(log^3(n))-bit size labels that can be seen as an efficient mechanism for verifying double-dominators. Expand
All-Pairs 2-Reachability in O(n^w log n) Time
This paper presents an algorithm that computes 2-reachability information for all pairs of vertices in O(n^w log n) time, and shows that the running time of all-pairs 2- reachability is only within a log factor of transitive closure. Expand
Connectivity Oracles for Planar Graphs
An implication of Pǎtrascu and Thorup's lower bound on predecessor search is that no d-failure connectivity oracle (even on trees) can beat pred(d,n) query time. Expand
Single source distance oracle for planar digraphs avoiding a failed node or link
The all-pairs version of this problem is addressed and a data structure with O(n ∼n polylog n) preprocessing time and space which guarantees O(√n poly log n) query time is presented. Expand
Efficient Vertex-Label Distance Oracles for Planar Graphs
It is shown how to preprocess a directed planar graph with vertex labels and arc lengths into a data structure that answers queries of the following form. Expand
Connectivity oracles for failure prone graphs
This paper presents the first efficient connectivity oracle for graphs susceptible to vertex failures, and shows there is an ~O(m)-space oracle that processes any set of d failed edges in O(d2 log log n) time and, thereafter, answers connectivity queries in O-log log n time. Expand
2-Vertex Connectivity in Directed Graphs
This paper shows how to build in linear time an O(n)-space data structure, which can answer in constant time queries on whether any two vertices are 2-vertex-connected, and can produce a “witness” of this property when two query vertices v and w are not 2- Vertex connectivity. Expand
Compact oracles for reachability and approximate distances in planar digraphs
  • M. Thorup
  • Mathematics, Computer Science
  • JACM
  • 2004
It is shown that a planar digraph can be preprocessed in near-linear time, producing a near-linear space oracle that can answer reachability queries in constant time. The oracle can be distributed asExpand
Single Source -- All Sinks Max Flows in Planar Digraphs
No algorithm which could solve the all-pairs max st-How values problem faster than the time of 8(n2) max-How computations for every planar digraph was known is known. Expand
Strong Connectivity in Directed Graphs under Failures, with Applications
With the help of the data structures built, this work can design efficient algorithms for several other connectivity problems on digraphs and obtain in linear time a strongly connected spanning subgraph of G with O(n) edges that maintains the 1-connectivity cuts of G and the decompositions induced by those cuts. Expand