20,604 research outputs found
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
Proof of the Labastida-Marino-Ooguri-Vafa Conjecture
Based on large N Chern-Simons/topological string duality, in a series of
papers, J.M.F. Labastida, M. Marino, H. Ooguri and C. Vafa conjectured certain
remarkable new algebraic structure of link invariants and the existence of
infinite series of new integer invariants. In this paper, we provide a proof of
this conjecture. Moreover, we also show these new integer invariants vanish at
large genera.Comment: 57pages, typos corrected, add some detail
Holographic entanglement entropy in general holographic superconductor models
We study the entanglement entropy of general holographic dual models both in
AdS soliton and AdS black hole backgrounds with full backreaction. We find that
the entanglement entropy is a good probe to explore the properties of the
holographic superconductors and provides richer physics in the phase
transition. We obtain the effects of the scalar mass, model parameter and
backreaction on the entropy, and argue that the jump of the entanglement
entropy may be a quite general feature for the first order phase transition. In
strong contrast to the insulator/superconductor system, we note that the
backreaction coupled with the scalar mass can not be used to trigger the first
order phase transition if the model parameter is below its bottom bound in the
metal/superconductor system.Comment: 14 pages, 6 figures. arXiv admin note: text overlap with
arXiv:1203.6620 by other author
Testing Small Set Expansion in General Graphs
We consider the problem of testing small set expansion for general graphs. A
graph is a -expander if every subset of volume at most has
conductance at least . Small set expansion has recently received
significant attention due to its close connection to the unique games
conjecture, the local graph partitioning algorithms and locally testable codes.
We give testers with two-sided error and one-sided error in the adjacency
list model that allows degree and neighbor queries to the oracle of the input
graph. The testers take as input an -vertex graph , a volume bound ,
an expansion bound and a distance parameter . For the
two-sided error tester, with probability at least , it accepts the graph
if it is a -expander and rejects the graph if it is -far
from any -expander, where and
. The
query complexity and running time of the tester are
, where is the number of
edges of the graph. For the one-sided error tester, it accepts every
-expander, and with probability at least , rejects every graph
that is -far from -expander, where
and for any . The query
complexity and running time of this tester are
.
We also give a two-sided error tester with smaller gap between and
in the rotation map model that allows (neighbor, index) queries and
degree queries.Comment: 23 pages; STACS 201
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