48,592 research outputs found
Sparse Regular Expression Matching
We present the first algorithm for regular expression matching that can take
advantage of sparsity in the input instance. Our main result is a new algorithm
that solves regular expression matching in time, where is the number of positions in
the regular expression, is the length of the string, and is the
\emph{density} of the instance, defined as the total number of active states in
a simulation of the position automaton. This measure is a lower bound on the
total number of active states in simulations of all classic polynomial sized
finite automata. Our bound improves the best known bounds for regular
expression matching by almost a linear factor in the density of the problem.
The key component in the result is a novel linear space representation of the
position automaton that supports state-set transition computation in
near-linear time in the size of the input and output state sets
Dense 3D Face Correspondence
We present an algorithm that automatically establishes dense correspondences
between a large number of 3D faces. Starting from automatically detected sparse
correspondences on the outer boundary of 3D faces, the algorithm triangulates
existing correspondences and expands them iteratively by matching points of
distinctive surface curvature along the triangle edges. After exhausting
keypoint matches, further correspondences are established by generating evenly
distributed points within triangles by evolving level set geodesic curves from
the centroids of large triangles. A deformable model (K3DM) is constructed from
the dense corresponded faces and an algorithm is proposed for morphing the K3DM
to fit unseen faces. This algorithm iterates between rigid alignment of an
unseen face followed by regularized morphing of the deformable model. We have
extensively evaluated the proposed algorithms on synthetic data and real 3D
faces from the FRGCv2, Bosphorus, BU3DFE and UND Ear databases using
quantitative and qualitative benchmarks. Our algorithm achieved dense
correspondences with a mean localisation error of 1.28mm on synthetic faces and
detected anthropometric landmarks on unseen real faces from the FRGCv2
database with 3mm precision. Furthermore, our deformable model fitting
algorithm achieved 98.5% face recognition accuracy on the FRGCv2 and 98.6% on
Bosphorus database. Our dense model is also able to generalize to unseen
datasets.Comment: 24 Pages, 12 Figures, 6 Tables and 3 Algorithm
Percolation on sparse random graphs with given degree sequence
We study the two most common types of percolation process on a sparse random
graph with a given degree sequence. Namely, we examine first a bond percolation
process where the edges of the graph are retained with probability p and
afterwards we focus on site percolation where the vertices are retained with
probability p. We establish critical values for p above which a giant component
emerges in both cases. Moreover, we show that in fact these coincide. As a
special case, our results apply to power law random graphs. We obtain rigorous
proofs for formulas derived by several physicists for such graphs.Comment: 20 page
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