48 research outputs found
Faster Deterministic Fully-Dynamic Graph Connectivity
We give new deterministic bounds for fully-dynamic graph connectivity. Our
data structure supports updates (edge insertions/deletions) in
amortized time and connectivity queries in worst-case time, where is the number of vertices of the
graph. This improves the deterministic data structures of Holm, de Lichtenberg,
and Thorup (STOC 1998, J.ACM 2001) and Thorup (STOC 2000) which both have
amortized update time and worst-case query
time. Our model of computation is the same as that of Thorup, i.e., a pointer
machine with standard instructions.Comment: To appear at SODA 2013. 19 pages, 1 figur
Optimal decremental connectivity in planar graphs
We show an algorithm for dynamic maintenance of connectivity information in
an undirected planar graph subject to edge deletions. Our algorithm may answer
connectivity queries of the form `Are vertices and connected with a
path?' in constant time. The queries can be intermixed with any sequence of
edge deletions, and the algorithm handles all updates in time. This
results improves over previously known time algorithm
Faster Worst Case Deterministic Dynamic Connectivity
We present a deterministic dynamic connectivity data structure for undirected
graphs with worst case update time and constant query time. This improves on the previous best
deterministic worst case algorithm of Frederickson (STOC 1983) and Eppstein
Galil, Italiano, and Nissenzweig (J. ACM 1997), which had update time
. All other algorithms for dynamic connectivity are either
randomized (Monte Carlo) or have only amortized performance guarantees
Reliable Hubs for Partially-Dynamic All-Pairs Shortest Paths in Directed Graphs
We give new partially-dynamic algorithms for the all-pairs shortest paths problem in weighted directed graphs. Most importantly, we give a new deterministic incremental algorithm for the problem that handles updates in O~(mn^(4/3) log{W}/epsilon) total time (where the edge weights are from [1,W]) and explicitly maintains a (1+epsilon)-approximate distance matrix. For a fixed epsilon>0, this is the first deterministic partially dynamic algorithm for all-pairs shortest paths in directed graphs, whose update time is o(n^2) regardless of the number of edges. Furthermore, we also show how to improve the state-of-the-art partially dynamic randomized algorithms for all-pairs shortest paths [Baswana et al. STOC\u2702, Bernstein STOC\u2713] from Monte Carlo randomized to Las Vegas randomized without increasing the running time bounds (with respect to the O~(*) notation).
Our results are obtained by giving new algorithms for the problem of dynamically maintaining hubs, that is a set of O~(n/d) vertices which hit a shortest path between each pair of vertices, provided it has hop-length Omega(d). We give new subquadratic deterministic and Las Vegas algorithms for maintenance of hubs under either edge insertions or deletions
Simple Greedy 2-Approximation Algorithm for the Maximum Genus of a Graph
The maximum genus gamma_M(G) of a graph G is the largest genus of an orientable surface into which G has a cellular embedding. Combinatorially, it coincides with the maximum number of disjoint pairs of adjacent edges of G whose removal results in a connected spanning subgraph of G. In this paper we describe a greedy 2-approximation algorithm for maximum genus by proving that removing pairs of adjacent edges from G arbitrarily while retaining connectedness leads to at least gamma_M(G)/2 pairs of edges removed. As a consequence of our approach we also obtain a 2-approximate counterpart of Xuong\u27s combinatorial characterisation of maximum genus
Faster Fully-Dynamic Minimum Spanning Forest
We give a new data structure for the fully-dynamic minimum spanning forest
problem in simple graphs. Edge updates are supported in
amortized time per operation, improving the amortized bound of
Holm et al. (STOC'98, JACM'01). We assume the Word-RAM model with standard
instructions.Comment: 13 pages, 2 figure
Incremental and Fully Dynamic Subgraph Connectivity For Emergency Planning
During the last 10 years it has become popular to study dynamic graph problems in a emergency planning or sensitivity setting: Instead of considering the general fully dynamic problem, we only have to process a single batch update of size d; after the update we have to answer queries.
In this paper, we consider the dynamic subgraph connectivity problem with sensitivity d: We are given a graph of which some vertices are activated and some are deactivated. After that we get a single update in which the states of up to vertices are changed. Then we get a sequence of connectivity queries in the subgraph of activated vertices.
We present the first fully dynamic algorithm for this problem which has an update and query time only slightly worse than the best decremental algorithm. In addition, we present the first incremental algorithm which is tight with respect to the best known conditional lower bound; moreover, the algorithm is simple and we believe it is implementable and efficient in practice
Fast Dynamic Graph Algorithms for Parameterized Problems
Fully dynamic graph is a data structure that (1) supports edge insertions and
deletions and (2) answers problem specific queries. The time complexity of (1)
and (2) are referred to as the update time and the query time respectively.
There are many researches on dynamic graphs whose update time and query time
are , that is, sublinear in the graph size. However, almost all such
researches are for problems in P. In this paper, we investigate dynamic graphs
for NP-hard problems exploiting the notion of fixed parameter tractability
(FPT).
We give dynamic graphs for Vertex Cover and Cluster Vertex Deletion
parameterized by the solution size . These dynamic graphs achieve almost the
best possible update time and the query time
, where is the time complexity of any static
graph algorithm for the problems. We obtain these results by dynamically
maintaining an approximate solution which can be used to construct a small
problem kernel. Exploiting the dynamic graph for Cluster Vertex Deletion, as a
corollary, we obtain a quasilinear-time (polynomial) kernelization algorithm
for Cluster Vertex Deletion. Until now, only quadratic time kernelization
algorithms are known for this problem.
We also give a dynamic graph for Chromatic Number parameterized by the
solution size of Cluster Vertex Deletion, and a dynamic graph for
bounded-degree Feedback Vertex Set parameterized by the solution size. Assuming
the parameter is a constant, each dynamic graph can be updated in
time and can compute a solution in time. These results are obtained by
another approach.Comment: SWAT 2014 to appea