5,304 research outputs found
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
Restricted Space Algorithms for Isomorphism on Bounded Treewidth Graphs
The Graph Isomorphism problem restricted to graphs of bounded treewidth or
bounded tree distance width are known to be solvable in polynomial time
[Bod90],[YBFT99]. We give restricted space algorithms for these problems
proving the following results: - Isomorphism for bounded tree distance width
graphs is in L and thus complete for the class. We also show that for this kind
of graphs a canon can be computed within logspace. - For bounded treewidth
graphs, when both input graphs are given together with a tree decomposition,
the problem of whether there is an isomorphism which respects the
decompositions (i.e. considering only isomorphisms mapping bags in one
decomposition blockwise onto bags in the other decomposition) is in L. - For
bounded treewidth graphs, when one of the input graphs is given with a tree
decomposition the isomorphism problem is in LogCFL. - As a corollary the
isomorphism problem for bounded treewidth graphs is in LogCFL. This improves
the known TC1 upper bound for the problem given by Grohe and Verbitsky
[GroVer06].Comment: STACS conference 2010, 12 page
Faster Separators for Shallow Minor-Free Graphs via Dynamic Approximate Distance Oracles
Plotkin, Rao, and Smith (SODA'97) showed that any graph with edges and
vertices that excludes as a depth -minor has a
separator of size and that such a separator can be
found in time. A time bound of for
any constant was later given (W., FOCS'11) which is an
improvement for non-sparse graphs. We give three new algorithms. The first has
the same separator size and running time O(\mbox{poly}(h)\ell
m^{1+\epsilon}). This is a significant improvement for small and .
If for an arbitrarily small chosen constant
, we get a time bound of O(\mbox{poly}(h)\ell n^{1+\epsilon}).
The second algorithm achieves the same separator size (with a slightly larger
polynomial dependency on ) and running time O(\mbox{poly}(h)(\sqrt\ell
n^{1+\epsilon} + n^{2+\epsilon}/\ell^{3/2})) when . Our third algorithm has running time
O(\mbox{poly}(h)\sqrt\ell n^{1+\epsilon}) when . It finds a separator of size O(n/\ell) + \tilde
O(\mbox{poly}(h)\ell\sqrt n) which is no worse than previous bounds when
is fixed and . A main tool in obtaining our results
is a novel application of a decremental approximate distance oracle of Roditty
and Zwick.Comment: 16 pages. Full version of the paper that appeared at ICALP'14. Minor
fixes regarding the time bounds such that these bounds hold also for
non-sparse graph
Minimum Cycle Basis and All-Pairs Min Cut of a Planar Graph in Subquadratic Time
A minimum cycle basis of a weighted undirected graph is a basis of the
cycle space of such that the total weight of the cycles in this basis is
minimized. If is a planar graph with non-negative edge weights, such a
basis can be found in time and space, where is the size of . We
show that this is optimal if an explicit representation of the basis is
required. We then present an time and space
algorithm that computes a minimum cycle basis \emph{implicitly}. From this
result, we obtain an output-sensitive algorithm that explicitly computes a
minimum cycle basis in time and space,
where is the total size (number of edges and vertices) of the cycles in the
basis. These bounds reduce to and ,
respectively, when is unweighted. We get similar results for the all-pairs
min cut problem since it is dual equivalent to the minimum cycle basis problem
for planar graphs. We also obtain time and
space algorithms for finding, respectively, the weight vector and a Gomory-Hu
tree of . The previous best time and space bound for these two problems was
quadratic. From our Gomory-Hu tree algorithm, we obtain the following result:
with time and space for preprocessing, the
weight of a min cut between any two given vertices of can be reported in
constant time. Previously, such an oracle required quadratic time and space for
preprocessing. The oracle can also be extended to report the actual cut in time
proportional to its size
Join-Reachability Problems in Directed Graphs
For a given collection G of directed graphs we define the join-reachability
graph of G, denoted by J(G), as the directed graph that, for any pair of
vertices a and b, contains a path from a to b if and only if such a path exists
in all graphs of G. Our goal is to compute an efficient representation of J(G).
In particular, we consider two versions of this problem. In the explicit
version we wish to construct the smallest join-reachability graph for G. In the
implicit version we wish to build an efficient data structure (in terms of
space and query time) such that we can report fast the set of vertices that
reach a query vertex in all graphs of G. This problem is related to the
well-studied reachability problem and is motivated by emerging applications of
graph-structured databases and graph algorithms. We consider the construction
of join-reachability structures for two graphs and develop techniques that can
be applied to both the explicit and the implicit problem. First we present
optimal and near-optimal structures for paths and trees. Then, based on these
results, we provide efficient structures for planar graphs and general directed
graphs
Fine-To-Coarse Global Registration of RGB-D Scans
RGB-D scanning of indoor environments is important for many applications,
including real estate, interior design, and virtual reality. However, it is
still challenging to register RGB-D images from a hand-held camera over a long
video sequence into a globally consistent 3D model. Current methods often can
lose tracking or drift and thus fail to reconstruct salient structures in large
environments (e.g., parallel walls in different rooms). To address this
problem, we propose a "fine-to-coarse" global registration algorithm that
leverages robust registrations at finer scales to seed detection and
enforcement of new correspondence and structural constraints at coarser scales.
To test global registration algorithms, we provide a benchmark with 10,401
manually-clicked point correspondences in 25 scenes from the SUN3D dataset.
During experiments with this benchmark, we find that our fine-to-coarse
algorithm registers long RGB-D sequences better than previous methods
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