130 research outputs found
Cache-Oblivious Peeling of Random Hypergraphs
The computation of a peeling order in a randomly generated hypergraph is the
most time-consuming step in a number of constructions, such as perfect hashing
schemes, random -SAT solvers, error-correcting codes, and approximate set
encodings. While there exists a straightforward linear time algorithm, its poor
I/O performance makes it impractical for hypergraphs whose size exceeds the
available internal memory.
We show how to reduce the computation of a peeling order to a small number of
sequential scans and sorts, and analyze its I/O complexity in the
cache-oblivious model. The resulting algorithm requires
I/Os and time to peel a random hypergraph with edges.
We experimentally evaluate the performance of our implementation of this
algorithm in a real-world scenario by using the construction of minimal perfect
hash functions (MPHF) as our test case: our algorithm builds a MPHF of
billion keys in less than hours on a single machine. The resulting data
structure is both more space-efficient and faster than that obtained with the
current state-of-the-art MPHF construction for large-scale key sets
A sharp threshold for random graphs with a monochromatic triangle in every edge coloring
Let be the set of all finite graphs with the Ramsey property that
every coloring of the edges of by two colors yields a monochromatic
triangle. In this paper we establish a sharp threshold for random graphs with
this property. Let be the random graph on vertices with edge
probability . We prove that there exists a function with
, as tends to infinity
Pr[G(n,(1-\eps)\hat c/\sqrt{n}) \in \R ] \to 0 and Pr [ G(n,(1+\eps)\hat
c/\sqrt{n}) \in \R ] \to 1. A crucial tool that is used in the proof and is
of independent interest is a generalization of Szemer\'edi's Regularity Lemma
to a certain hypergraph setting.Comment: 101 pages, Final version - to appear in Memoirs of the A.M.
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
Wickets in 3-uniform Hypergraphs
In these notes, we consider a Tur\'an-type problem in hypergraphs. What is
the maximum number of edges if we forbid a subgraph? Let be a
3-uniform linear hypergraph, i.e. any two edges have at most one vertex common.
A special hypergraph, called {\em wicket}, is formed by three rows and two
columns of a point matrix. We describe two linear hypergraphs --
both containing a wicket -- that if we forbid either of them in ,
then the hypergraph is sparse, and the number of its edges is . This
proves a conjecture of Gy\'arf\'as and S\'ark\"ozy
- …