Planning for Fast Connectivity Updates

  title={Planning for Fast Connectivity Updates},
  author={Mihai Patrascu and Mikkel Thorup},
  journal={48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)},
  • M. Patrascu, M. Thorup
  • Published 21 October 2007
  • Computer Science
  • 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)
Understanding how a single edge deletion can affect the connectivity of a graph amounts to finding the graph bridges. But when faced with d. > l deletions, can we establish as easily how the connectivity changes? When planning for an emergency, we want to understand the structure of our network ahead of time, and respond swiftly when an emergency actually happens. We describe a linear-space representation of graphs which enables us to determine how a batch of edge updates can impact the graph… 
Dynamic Connectivity: Connecting to Networks and Geometry
A data structure supporting vertex updates in O~(m^{2/3}) amortized time is described, where m denotes the number of edges in the graph, and it is shown how to obtain sublinear update bounds for virtually all families of geometric objects which allow sublinear-time range queries.
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.
Incremental and Fully Dynamic Subgraph Connectivity For Emergency Planning
The first fully dynamic algorithm for the dynamic subgraph connectivity problem with sensitivity d is presented, which has an update and query time only slightly worse than the best decremental algorithm.
Faster Randomized Worst-Case Update Time for Dynamic Subgraph Connectivity
In the general undirected graph, a randomized data structure is proposed, which has \(\widetilde{O}(m^{3/4})\) worst-case update time, which answers the queries on connectivity between any two active vertices in the subgraph of G induced by S.
Maintaining alterable planar embeddings of dynamic graphs
A data structure for maintaining a planar embedding of a dynamic plane graph which supports edge deleting, edge insertion, and query, and a new lower bound for dynamic planarity testing is presented, namely Ω(lg(n) for the slowest operation.
New Data Structures for Subgraph Connectivity
  • Ran Duan
  • Mathematics, Computer Science
  • 2010
The first subgraph connectivity structure with worst-case sublinear time bounds for both updates and queries is given, which improves the structure introduced by [Chan, Patrascu, Roditty, FOCS'08] that takes O(m4/3) space.
Incremental Strong Connectivity and 2-Connectivity in Directed Graphs
A conditional lower bound is given that provides evidence that the new incremental algorithms for maintaining data structures that represent all connectivity cuts of size one in directed graphs (digraphs) may be tight up to sub-polynomial factors.
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.
Optimal Labeling for Connectivity Checking in Planar Networks with Obstacles
We consider the problem of determining in a planar graph G whether two vertices x and y are linked by a path that avoids a set X of vertices and a set F of edges. We attach labels to vertices in such
Faster Worst-Case Update Time for Dynamic Subgraph Connectivity
A structure of linear space, with worst-case update time $\widetilde{O}(m^{3/4})$.


Worst-case update times for fully-dynamic all-pairs shortest paths
The first solution to the fully-dynamic all pairs shortest path problem where every update is faster than a recomputation from scratch in Ω(n) time is presented, for a directed graph with arbitrary non-negative edge weights.
Randomized dynamic graph algorithms with polylogarithmic time per operation
This paper presents the first fully dynamic algorithms that maintain connectivity, bipartiteness, and approximate minimum spanning trees in polylogarithmic time per edge insertion or deletion using a new dynamic technique that combines a novel graph decomposition with randomization.
Sparsification-a technique for speeding up dynamic graph algorithms
The authors provide data structures that maintain a graph as edges are inserted and deleted, and keep track of the following properties: minimum spanning forests, best swap, graph connectivity, and
Improved Distance Oracles for Avoiding Link-Failure
We consider the problem of preprocessing an edge-weighted directed graph to answer queries that ask for the shortest path from any given vertex to another avoiding a failed link. We present two
Expander flows, geometric embeddings and graph partitioning
An interesting and natural “approximate certificate” for a graph's expansion, which involves embedding an n-node expander in it with appropriate dilation and congestion, is described.
Near-optimal fully-dynamic graph connectivity
Near-optimal bounds for fullydynamic graph connectivity which is the most basic nontrivial fully-d dynamic graph problem are presented and some comparatively trivial observations are made improving some deterministic bounds.
Oracles for distances avoiding a link-failure
For a directed graph <i>G</i> we consider queries of the form: "What is the shortest path distance from vertex <i>x</i> to vertex <i>y</i> in <i>G</i> avoiding a failed link (<i>u, v</i>), and what
Estimating the sortedness of a data stream
It is conjecture that any deterministic (1 + ε) approximation algorithm for LIS requires Ω (√n) space, and a lower bound of Ω(n) is proved for a restricted yet natural class of deterministic algorithms.
Logarithmic Lower Bounds in the Cell-Probe Model
A new technique for proving cell-probe lower bounds on dynamic data structures is developed, which enables an amortized randomized $\Omega(\lg n)$ lower bound per operation for several data structural problems on $n$ elements, including partial sums, dynamic connectivity among disjoint paths, and several other dynamic graph problems (by simple reductions).
Data Structures for On-Line Updating of Minimum Spanning Trees, with Applications
Data structures are presented for the problem of maintaining a minimum spanning tree on-line under the operation of updating the cost of some edge in the graph. For the case of a general graph, mai...