330 research outputs found
The Sketching Complexity of Graph and Hypergraph Counting
Subgraph counting is a fundamental primitive in graph processing, with
applications in social network analysis (e.g., estimating the clustering
coefficient of a graph), database processing and other areas. The space
complexity of subgraph counting has been studied extensively in the literature,
but many natural settings are still not well understood. In this paper we
revisit the subgraph (and hypergraph) counting problem in the sketching model,
where the algorithm's state as it processes a stream of updates to the graph is
a linear function of the stream. This model has recently received a lot of
attention in the literature, and has become a standard model for solving
dynamic graph streaming problems.
In this paper we give a tight bound on the sketching complexity of counting
the number of occurrences of a small subgraph in a bounded degree graph
presented as a stream of edge updates. Specifically, we show that the space
complexity of the problem is governed by the fractional vertex cover number of
the graph . Our subgraph counting algorithm implements a natural vertex
sampling approach, with sampling probabilities governed by the vertex cover of
. Our main technical contribution lies in a new set of Fourier analytic
tools that we develop to analyze multiplayer communication protocols in the
simultaneous communication model, allowing us to prove a tight lower bound. We
believe that our techniques are likely to find applications in other settings.
Besides giving tight bounds for all graphs , both our algorithm and lower
bounds extend to the hypergraph setting, albeit with some loss in space
complexity
Activity recognition from videos with parallel hypergraph matching on GPUs
In this paper, we propose a method for activity recognition from videos based
on sparse local features and hypergraph matching. We benefit from special
properties of the temporal domain in the data to derive a sequential and fast
graph matching algorithm for GPUs.
Traditionally, graphs and hypergraphs are frequently used to recognize
complex and often non-rigid patterns in computer vision, either through graph
matching or point-set matching with graphs. Most formulations resort to the
minimization of a difficult discrete energy function mixing geometric or
structural terms with data attached terms involving appearance features.
Traditional methods solve this minimization problem approximately, for instance
with spectral techniques.
In this work, instead of solving the problem approximatively, the exact
solution for the optimal assignment is calculated in parallel on GPUs. The
graphical structure is simplified and regularized, which allows to derive an
efficient recursive minimization algorithm. The algorithm distributes
subproblems over the calculation units of a GPU, which solves them in parallel,
allowing the system to run faster than real-time on medium-end GPUs
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
Open Problems in (Hyper)Graph Decomposition
Large networks are useful in a wide range of applications. Sometimes problem
instances are composed of billions of entities. Decomposing and analyzing these
structures helps us gain new insights about our surroundings. Even if the final
application concerns a different problem (such as traversal, finding paths,
trees, and flows), decomposing large graphs is often an important subproblem
for complexity reduction or parallelization. This report is a summary of
discussions that happened at Dagstuhl seminar 23331 on "Recent Trends in Graph
Decomposition" and presents currently open problems and future directions in
the area of (hyper)graph decomposition
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Data Stream Algorithms for Large Graphs and High Dimensional Data
In contrast to the traditional random access memory computational model where the entire input is available in the working memory, the data stream model only provides sequential access to the input. The data stream model is a natural framework to handle large and dynamic data. In this model, we focus on designing algorithms that use sublinear memory and a small number of passes over the stream. Other desirable properties include fast update time, query time, and post processing time.
In this dissertation, we consider different problems in graph theory, combinatorial optimization, and high dimensional data processing.
The first part of this dissertation focuses on algorithms for graph theory and combinatorial optimization. We present new results for the problems of finding the densest subgraph, counting the number of triangles, finding max cut with bounded components, and finding the maximum set coverage.
The second part of this dissertation considers problems in high dimensional data streams. In this setting, each stream item consists of multiple coordinates corresponding to different attributes. We consider the problem of testing or learning about the relationships among the attributes, and the problem of finding heavy hitters in subsets of attributes
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