114 research outputs found
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
Choosing Colors for Geometric Graphs via Color Space Embeddings
Graph drawing research traditionally focuses on producing geometric
embeddings of graphs satisfying various aesthetic constraints. After the
geometric embedding is specified, there is an additional step that is often
overlooked or ignored: assigning display colors to the graph's vertices. We
study the additional aesthetic criterion of assigning distinct colors to
vertices of a geometric graph so that the colors assigned to adjacent vertices
are as different from one another as possible. We formulate this as a problem
involving perceptual metrics in color space and we develop algorithms for
solving this problem by embedding the graph in color space. We also present an
application of this work to a distributed load-balancing visualization problem.Comment: 12 pages, 4 figures. To appear at 14th Int. Symp. Graph Drawing, 200
Spanners for Geometric Intersection Graphs
Efficient algorithms are presented for constructing spanners in geometric
intersection graphs. For a unit ball graph in R^k, a (1+\epsilon)-spanner is
obtained using efficient partitioning of the space into hypercubes and solving
bichromatic closest pair problems. The spanner construction has almost
equivalent complexity to the construction of Euclidean minimum spanning trees.
The results are extended to arbitrary ball graphs with a sub-quadratic running
time.
For unit ball graphs, the spanners have a small separator decomposition which
can be used to obtain efficient algorithms for approximating proximity problems
like diameter and distance queries. The results on compressed quadtrees,
geometric graph separators, and diameter approximation might be of independent
interest.Comment: 16 pages, 5 figures, Late
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
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