27,414 research outputs found
Dimers, webs, and positroids
We study the dimer model for a planar bipartite graph N embedded in a disk,
with boundary vertices on the boundary of the disk. Counting dimer
configurations with specified boundary conditions gives a point in the totally
nonnegative Grassmannian. Considering pairing probabilities for the
double-dimer model gives rise to Grassmann analogues of Rhoades and Skandera's
Temperley-Lieb immanants. The same problem for the (probably novel)
triple-dimer model gives rise to the combinatorics of Kuperberg's webs and
Grassmann analogues of Pylyavskyy's web immanants. This draws a connection
between the square move of plabic graphs (or urban renewal of planar bipartite
graphs), and Kuperberg's square reduction of webs. Our results also suggest
that canonical-like bases might be applied to the dimer model.
We furthermore show that these functions on the Grassmannian are compatible
with restriction to positroid varieties. Namely, our construction gives bases
for the degree two and degree three components of the homogeneous coordinate
ring of a positroid variety that are compatible with the cyclic group action.Comment: 25 page
Mapping class group orbits of curves with self-intersections
We study mapping class group orbits of homotopy and isotopy classes of curves
with self-intersections. We exhibit the asymptotics of the number of such
orbits of curves with a bounded number of self-intersections, as the complexity
of the surface tends to infinity. We also consider the minimal genus of a
subsurface that contains the curve. We determine the asymptotic number of
orbits of curves with a fixed minimal genus and a bounded self-intersection
number, as the complexity of the surface tends to infinity. As a corollary of
our methods, we obtain that most curves that are homotopic are also isotopic.
Furthermore, using a theorem by Basmajian, we get a bound on the number of
mapping class group orbits on a given a hyperbolic surface that can contain
short curves. For a fixed length, this bound is polynomial in the signature of
the surface. The arguments we use are based on counting embeddings of ribbon
graphs.Comment: 16 pages, 1 figure, generalized main resul
Distributed Triangle Counting in the Graphulo Matrix Math Library
Triangle counting is a key algorithm for large graph analysis. The Graphulo
library provides a framework for implementing graph algorithms on the Apache
Accumulo distributed database. In this work we adapt two algorithms for
counting triangles, one that uses the adjacency matrix and another that also
uses the incidence matrix, to the Graphulo library for server-side processing
inside Accumulo. Cloud-based experiments show a similar performance profile for
these different approaches on the family of power law Graph500 graphs, for
which data skew increasingly bottlenecks. These results motivate the design of
skew-aware hybrid algorithms that we propose for future work.Comment: Honorable mention in the 2017 IEEE HPEC's Graph Challeng
MSPKmerCounter: A Fast and Memory Efficient Approach for K-mer Counting
A major challenge in next-generation genome sequencing (NGS) is to assemble
massive overlapping short reads that are randomly sampled from DNA fragments.
To complete assembling, one needs to finish a fundamental task in many leading
assembly algorithms: counting the number of occurrences of k-mers (length-k
substrings in sequences). The counting results are critical for many components
in assembly (e.g. variants detection and read error correction). For large
genomes, the k-mer counting task can easily consume a huge amount of memory,
making it impossible for large-scale parallel assembly on commodity servers.
In this paper, we develop MSPKmerCounter, a disk-based approach, to
efficiently perform k-mer counting for large genomes using a small amount of
memory. Our approach is based on a novel technique called Minimum Substring
Partitioning (MSP). MSP breaks short reads into multiple disjoint partitions
such that each partition can be loaded into memory and processed individually.
By leveraging the overlaps among the k-mers derived from the same short read,
MSP can achieve astonishing compression ratio so that the I/O cost can be
significantly reduced. For the task of k-mer counting, MSPKmerCounter offers a
very fast and memory-efficient solution. Experiment results on large real-life
short reads data sets demonstrate that MSPKmerCounter can achieve better
overall performance than state-of-the-art k-mer counting approaches.
MSPKmerCounter is available at http://www.cs.ucsb.edu/~yangli/MSPKmerCounte
The Ising Partition Function: Zeros and Deterministic Approximation
We study the problem of approximating the partition function of the
ferromagnetic Ising model in graphs and hypergraphs. Our first result is a
deterministic approximation scheme (an FPTAS) for the partition function in
bounded degree graphs that is valid over the entire range of parameters
(the interaction) and (the external field), except for the case
(the "zero-field" case). A randomized algorithm (FPRAS)
for all graphs, and all , has long been known. Unlike most other
deterministic approximation algorithms for problems in statistical physics and
counting, our algorithm does not rely on the "decay of correlations" property.
Rather, we exploit and extend machinery developed recently by Barvinok, and
Patel and Regts, based on the location of the complex zeros of the partition
function, which can be seen as an algorithmic realization of the classical
Lee-Yang approach to phase transitions. Our approach extends to the more
general setting of the Ising model on hypergraphs of bounded degree and edge
size, where no previous algorithms (even randomized) were known for a wide
range of parameters. In order to achieve this extension, we establish a tight
version of the Lee-Yang theorem for the Ising model on hypergraphs, improving a
classical result of Suzuki and Fisher.Comment: clarified presentation of combinatorial arguments, added new results
on optimality of univariate Lee-Yang theorem
Two-dimensional gauge theories of the symmetric group S(n) and branched n-coverings of Riemann surfaces in the large-n limit
Branched n-coverings of Riemann surfaces are described by a 2d lattice gauge
theory of the symmetric group S(n) defined on a cell discretization of the
surface. We study the theory in the large-n limit, and we find a rich phase
diagram with first and second order transition lines. The various phases are
characterized by different connectivity properties of the covering surface. We
point out some interesting connections with the theory of random walks on group
manifolds and with random graph theory.Comment: Talk presented at the "Light-cone physics: particles and strings",
Trento, Italy, September 200
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