4,038 research outputs found
Dynamic Connectivity: Connecting to Networks and Geometry
Dynamic connectivity is a well-studied problem, but so far the most
compelling progress has been confined to the edge-update model: maintain an
understanding of connectivity in an undirected graph, subject to edge
insertions and deletions. In this paper, we study two more challenging, yet
equally fundamental problems.
Subgraph connectivity asks to maintain an understanding of connectivity under
vertex updates: updates can turn vertices on and off, and queries refer to the
subgraph induced by "on" vertices. (For instance, this is closer to
applications in networks of routers, where node faults may occur.)
We describe a data structure supporting vertex updates in O (m^{2/3})
amortized time, where m denotes the number of edges in the graph. This greatly
improves over the previous result [Chan, STOC'02], which required fast matrix
multiplication and had an update time of O(m^0.94). The new data structure is
also simpler.
Geometric connectivity asks to maintain a dynamic set of n geometric objects,
and query connectivity in their intersection graph. (For instance, the
intersection graph of balls describes connectivity in a network of sensors with
bounded transmission radius.)
Previously, nontrivial fully dynamic results were known only for special
cases like axis-parallel line segments and rectangles. We provide similarly
improved update times, O (n^{2/3}), for these special cases. Moreover, we show
how to obtain sublinear update bounds for virtually all families of geometric
objects which allow sublinear-time range queries, such as arbitrary 2D line
segments, d-dimensional simplices, and d-dimensional balls.Comment: Full version of a paper to appear in FOCS 200
Dynamic Graph Stream Algorithms in Space
In this paper we study graph problems in dynamic streaming model, where the
input is defined by a sequence of edge insertions and deletions. As many
natural problems require space, where is the number of
vertices, existing works mainly focused on designing space
algorithms. Although sublinear in the number of edges for dense graphs, it
could still be too large for many applications (e.g. is huge or the graph
is sparse). In this work, we give single-pass algorithms beating this space
barrier for two classes of problems.
We present space algorithms for estimating the number of connected
components with additive error and
-approximating the weight of minimum spanning tree, for any
small constant . The latter improves previous
space algorithm given by Ahn et al. (SODA 2012) for connected graphs with
bounded edge weights.
We initiate the study of approximate graph property testing in the dynamic
streaming model, where we want to distinguish graphs satisfying the property
from graphs that are -far from having the property. We consider
the problem of testing -edge connectivity, -vertex connectivity,
cycle-freeness and bipartiteness (of planar graphs), for which, we provide
algorithms using roughly space, which is
for any constant .
To complement our algorithms, we present space
lower bounds for these problems, which show that such a dependence on
is necessary.Comment: ICALP 201
Diameter and Treewidth in Minor-Closed Graph Families
It is known that any planar graph with diameter D has treewidth O(D), and
this fact has been used as the basis for several planar graph algorithms. We
investigate the extent to which similar relations hold in other graph families.
We show that treewidth is bounded by a function of the diameter in a
minor-closed family, if and only if some apex graph does not belong to the
family. In particular, the O(D) bound above can be extended to bounded-genus
graphs. As a consequence, we extend several approximation algorithms and exact
subgraph isomorphism algorithms from planar graphs to other graph families.Comment: 15 pages, 12 figure
Worm Monte Carlo study of the honeycomb-lattice loop model
We present a Markov-chain Monte Carlo algorithm of "worm"type that correctly
simulates the O(n) loop model on any (finite and connected) bipartite cubic
graph, for any real n>0, and any edge weight, including the fully-packed limit
of infinite edge weight. Furthermore, we prove rigorously that the algorithm is
ergodic and has the correct stationary distribution. We emphasize that by using
known exact mappings when n=2, this algorithm can be used to simulate a number
of zero-temperature Potts antiferromagnets for which the Wang-Swendsen-Kotecky
cluster algorithm is non-ergodic, including the 3-state model on the
kagome-lattice and the 4-state model on the triangular-lattice. We then use
this worm algorithm to perform a systematic study of the honeycomb-lattice loop
model as a function of n<2, on the critical line and in the densely-packed and
fully-packed phases. By comparing our numerical results with Coulomb gas
theory, we identify the exact scaling exponents governing some fundamental
geometric and dynamic observables. In particular, we show that for all n<2, the
scaling of a certain return time in the worm dynamics is governed by the
magnetic dimension of the loop model, thus providing a concrete dynamical
interpretation of this exponent. The case n>2 is also considered, and we
confirm the existence of a phase transition in the 3-state Potts universality
class that was recently observed via numerical transfer matrix calculations.Comment: 33 pages, 12 figure
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