138,252 research outputs found
Sparse Fault-Tolerant BFS Trees
This paper addresses the problem of designing a sparse {\em fault-tolerant}
BFS tree, or {\em FT-BFS tree} for short, namely, a sparse subgraph of the
given network such that subsequent to the failure of a single edge or
vertex, the surviving part of still contains a BFS spanning tree for
(the surviving part of) . Our main results are as follows. We present an
algorithm that for every -vertex graph and source node constructs a
(single edge failure) FT-BFS tree rooted at with O(n \cdot
\min\{\Depth(s), \sqrt{n}\}) edges, where \Depth(s) is the depth of the BFS
tree rooted at . This result is complemented by a matching lower bound,
showing that there exist -vertex graphs with a source node for which any
edge (or vertex) FT-BFS tree rooted at has edges. We then
consider {\em fault-tolerant multi-source BFS trees}, or {\em FT-MBFS trees}
for short, aiming to provide (following a failure) a BFS tree rooted at each
source for some subset of sources . Again, tight bounds
are provided, showing that there exists a poly-time algorithm that for every
-vertex graph and source set of size constructs a
(single failure) FT-MBFS tree from each source , with
edges, and on the other hand there exist
-vertex graphs with source sets of cardinality , on
which any FT-MBFS tree from has edges.
Finally, we propose an approximation algorithm for constructing
FT-BFS and FT-MBFS structures. The latter is complemented by a hardness result
stating that there exists no approximation algorithm for these
problems under standard complexity assumptions
Analysis of Crowdsourced Sampling Strategies for HodgeRank with Sparse Random Graphs
Crowdsourcing platforms are now extensively used for conducting subjective
pairwise comparison studies. In this setting, a pairwise comparison dataset is
typically gathered via random sampling, either \emph{with} or \emph{without}
replacement. In this paper, we use tools from random graph theory to analyze
these two random sampling methods for the HodgeRank estimator. Using the
Fiedler value of the graph as a measurement for estimator stability
(informativeness), we provide a new estimate of the Fiedler value for these two
random graph models. In the asymptotic limit as the number of vertices tends to
infinity, we prove the validity of the estimate. Based on our findings, for a
small number of items to be compared, we recommend a two-stage sampling
strategy where a greedy sampling method is used initially and random sampling
\emph{without} replacement is used in the second stage. When a large number of
items is to be compared, we recommend random sampling with replacement as this
is computationally inexpensive and trivially parallelizable. Experiments on
synthetic and real-world datasets support our analysis
Existentially Closed Models and Conservation Results in Bounded Arithmetic
We develop model-theoretic techniques to obtain conservation results for first order Bounded Arithmetic theories, based on a hierarchical version of the well-known notion of an existentially closed model. We focus on the classical Buss' theories Si2 and Ti2 and prove that they are ∀Σbi conservative over their inference rule counterparts, and ∃∀Σbi conservative over their parameter-free versions. A similar analysis of the Σbi-replacement scheme is also developed. The proof method is essentially the same for all the schemes we deal with and shows that these conservation results between schemes and inference rules do not depend on the specific combinatorial or arithmetical content of those schemes. We show that similar conservation results can be derived, in a very general setting, for every scheme enjoying some syntactical (or logical) properties common to both the induction and replacement schemes. Hence, previous conservation results for induction and replacement can be also obtained as corollaries of these more general results.Ministerio de Educación y Ciencia MTM2005-08658Junta de Andalucía TIC-13
Motivic homotopy theory of group scheme actions
We define an unstable equivariant motivic homotopy category for an algebraic
group over a Noetherian base scheme. We show that equivariant algebraic
-theory is representable in the resulting homotopy category. Additionally,
we establish homotopical purity and blow-up theorems for finite abelian groups.Comment: Final version, to appear in Journal of Topology. arXiv admin note:
text overlap with arXiv:1403.191
Exactly Solvable Balanced Tenable Urns with Random Entries via the Analytic Methodology
This paper develops an analytic theory for the study of some Polya urns with
random rules. The idea is to extend the isomorphism theorem in Flajolet et al.
(2006), which connects deterministic balanced urns to a differential system for
the generating function. The methodology is based upon adaptation of operators
and use of a weighted probability generating function. Systems of differential
equations are developed, and when they can be solved, they lead to
characterization of the exact distributions underlying the urn evolution. We
give a few illustrative examples.Comment: 23rd International Meeting on Probabilistic, Combinatorial, and
Asymptotic Methods for the Analysis of Algorithms (AofA'12), Montreal :
Canada (2012
The K-theory of filtered deformations of graded polynomial algebras
Recent discoveries make it possible to compute the K-theory of certain rings
from their cyclic homology and certain versions of their cdh-cohomology. We
extend the work of G. Corti\~nas et al. who calculated the K-theory of, in
addition to many other varieties, cones over smooth varieties, or equivalently
the K-theory of homogeneous polynomial rings. We focus on specific examples of
polynomial rings, which happen to be filtered deformations of homogeneous
polynomial rings. Along the way, as a secondary result, we will develop a
method for computing the periodic cyclic homology of a singular variety as well
as the negative cyclic homology when the cyclic homology of that variety is
known. Finally, we will apply these methods to extend the results of Michler
who computed the cyclic homology of hypersurfaces with isolated singularities.Comment: 66 pages, PhD Thesi
A generalization of Ohkawa's theorem
A theorem due to Ohkawa states that the collection of Bousfield equivalence
classes of spectra is a set. We extend this result to arbitrary combinatorial
model categories.Comment: 13 pages; consequences in motivic homotopy theory have been adde
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