864 research outputs found

### Online Steiner Tree with Deletions

In the online Steiner tree problem, the input is a set of vertices that
appear one-by-one, and we have to maintain a Steiner tree on the current set of
vertices. The cost of the tree is the total length of edges in the tree, and we
want this cost to be close to the cost of the optimal Steiner tree at all
points in time. If we are allowed to only add edges, a tight bound of
$\Theta(\log n)$ on the competitiveness is known. Recently it was shown that if
we can add one new edge and make one edge swap upon every vertex arrival, we
can maintain a constant-competitive tree online.
But what if the set of vertices sees both additions and deletions? Again, we
would like to obtain a low-cost Steiner tree with as few edge changes as
possible. The original paper of Imase and Waxman had also considered this
model, and it gave a greedy algorithm that maintained a constant-competitive
tree online, and made at most $O(n^{3/2})$ edge changes for the first $n$
requests. In this paper give the following two results.
Our first result is an online algorithm that maintains a Steiner tree only
under deletions: we start off with a set of vertices, and at each time one of
the vertices is removed from this set: our Steiner tree no longer has to span
this vertex. We give an algorithm that changes only a constant number of edges
upon each request, and maintains a constant-competitive tree at all times. Our
algorithm uses the primal-dual framework and a global charging argument to
carefully make these constant number of changes.
We then study the natural greedy algorithm proposed by Imase and Waxman that
maintains a constant-competitive Steiner tree in the fully-dynamic model (where
each request either adds or deletes a vertex). Our second result shows that
this algorithm makes only a constant number of changes per request in an
amortized sense.Comment: An extended abstract appears in the SODA 2014 conferenc

### Deformation and break-up of viscoelastic droplets Using Lattice Boltzmann Models

We investigate the break-up of Newtonian/viscoelastic droplets in a
viscoelastic/Newtonian matrix under the hydrodynamic conditions of a confined
shear flow. Our numerical approach is based on a combination of
Lattice-Boltzmann models (LBM) and Finite Difference (FD) schemes. LBM are used
to model two immiscible fluids with variable viscosity ratio (i.e. the ratio of
the droplet to matrix viscosity); FD schemes are used to model viscoelasticity,
and the kinetics of the polymers is introduced using constitutive equations for
viscoelastic fluids with finitely extensible non-linear elastic dumbbells with
Peterlin's closure (FENE-P). We study both strongly and weakly confined cases
to highlight the role of matrix and droplet viscoelasticity in changing the
droplet dynamics after the startup of a shear flow. Simulations provide easy
access to quantities such as droplet deformation and orientation and will be
used to quantitatively predict the critical Capillary number at which the
droplet breaks, the latter being strongly correlated to the formation of
multiple neckings at break-up. This study complements our previous
investigation on the role of droplet viscoelasticity (A. Gupta \& M.
Sbragaglia, {\it Phys. Rev. E} {\bf 90}, 023305 (2014)), and is here further
extended to the case of matrix viscoelasticity.Comment: 8 pages, 5 figures, IUTAM Symposium on Multiphase flows with phase
change: challenges and opportunities, Hyderabad, India 201

### Melting of a nonequilibrium vortex crystal in a fluid film with polymers : elastic versus fluid turbulence

We perform a direct numerical simulation (DNS) of the forced, incompressible
two-dimensional Navier-Stokes equation coupled with the FENE-P equations for
the polymer-conformation tensor. The forcing is such that, without polymers and
at low Reynolds numbers \mbox{Re}, the film attains a steady state that is a
square lattice of vortices and anti-vortices. We find that, as we increase the
Weissenberg number \mbox{Wi}, a sequence of nonequilibrium phase transitions
transforms this lattice, first to spatially distorted, but temporally steady,
crystals and then to a sequence of crystals that oscillate in time,
periodically, at low \mbox{Wi}, and quasiperiodically, for slightly larger
\mbox{Wi}. Finally, the system becomes disordered and displays spatiotemporal
chaos and elastic turbulence. We then obtain the nonequilibrium phase diagram
for this system, in the \mbox{Wi} - \Omega plane, where \Omega \propto
{\mbox{Re}}, and show that (a) the boundary between the crystalline and
turbulent phases has a complicated, fractal-type character and (b) the
Okubo-Weiss parameter $\Lambda$ provides us with a natural measure for
characterizing the phases and transitions in this diagram.Comment: 16 pages, 17 figure

### Effect of polymer-stress diffusion in the numerical simulation of elastic turbulence

Elastic turbulence is a chaotic regime that emerges in polymer solutions at
low Reynolds numbers. A common way to ensure stability in numerical simulations
of polymer solutions is to add artificially large polymer-stress diffusion. In
order to assess the accuracy of this approach in the elastic-turbulence regime,
we compare numerical simulations of the two-dimensional Oldroyd-B and FENE-P
models sustained by a cellular force with and without artificial diffusion. We
find that artificial diffusion can have a dramatic effect even on the
large-scale properties of the flow and we show some of the spurious phenomena
that may arise when artificial diffusion is used.Comment: 17 page

### How the Experts Algorithm Can Help Solve LPs Online

We consider the problem of solving packing/covering LPs online, when the
columns of the constraint matrix are presented in random order. This problem
has received much attention and the main focus is to figure out how large the
right-hand sides of the LPs have to be (compared to the entries on the
left-hand side of the constraints) to allow $(1+\epsilon)$-approximations
online. It is known that the right-hand sides have to be $\Omega(\epsilon^{-2}
\log m)$ times the left-hand sides, where $m$ is the number of constraints.
In this paper we give a primal-dual algorithm that achieve this bound for
mixed packing/covering LPs. Our algorithms construct dual solutions using a
regret-minimizing online learning algorithm in a black-box fashion, and use
them to construct primal solutions. The adversarial guarantee that holds for
the constructed duals helps us to take care of most of the correlations that
arise in the algorithm; the remaining correlations are handled via martingale
concentration and maximal inequalities. These ideas lead to conceptually simple
and modular algorithms, which we hope will be useful in other contexts.Comment: An extended abstract appears in the 22nd European Symposium on
Algorithms (ESA 2014

### A Lattice Boltzmann study of the effects of viscoelasticity on droplet formation in microfluidic cross-junctions

Based on mesoscale lattice Boltzmann (LB) numerical simulations, we
investigate the effects of viscoelasticity on the break-up of liquid threads in
microfluidic cross-junctions, where droplets are formed by focusing a liquid
thread of a dispersed (d) phase into another co-flowing continuous (c)
immiscible phase. Working at small Capillary numbers, we investigate the
effects of non-Newtonian phases in the transition from droplet formation at the
cross-junction (DCJ) to droplet formation downstream of the cross-junction (DC)
(Liu $\&$ Zhang, ${\it Phys. ~Fluids.}$ ${\bf 23}$, 082101 (2011)). We will
analyze cases with ${\it Droplet ~Viscoelasticity}$ (DV), where viscoelastic
properties are confined in the dispersed phase, as well as cases with ${\it
Matrix ~Viscoelasticity}$ (MV), where viscoelastic properties are confined in
the continuous phase. Moderate flow-rate ratios $Q \approx {\cal O}(1)$ of the
two phases are considered in the present study. Overall, we find that the
effects are more pronounced in the case with MV, where viscoelasticity is found
to influence the break-up point of the threads, which moves closer to the
cross-junction and stabilizes. This is attributed to an increase of the polymer
feedback stress forming in the corner flows, where the side channels of the
device meet the main channel. Quantitative predictions on the break-up point of
the threads are provided as a function of the Deborah number, i.e. the
dimensionless number measuring the importance of viscoelasticity with respect
to Capillary forces.Comment: 15 pages, 14 figures. This Work applies the Numerical Methodology
described in arXiv:1406.2686 to the Problem of Droplet Generation in
Microfluidic Cross Junctions. arXiv admin note: substantial text overlap with
arXiv:1508.0014

### Minimum d-dimensional arrangement with fixed points

In the Minimum $d$-Dimensional Arrangement Problem (d-dimAP) we are given a
graph with edge weights, and the goal is to find a 1-1 map of the vertices into
$\mathbb{Z}^d$ (for some fixed dimension $d\geq 1$) minimizing the total
weighted stretch of the edges. This problem arises in VLSI placement and chip
design.
Motivated by these applications, we consider a generalization of d-dimAP,
where the positions of some of the vertices (pins) is fixed and specified as
part of the input. We are asked to extend this partial map to a map of all the
vertices, again minimizing the weighted stretch of edges. This generalization,
which we refer to as d-dimAP+, arises naturally in these application domains
(since it can capture blocked-off parts of the board, or the requirement of
power-carrying pins to be in certain locations, etc.). Perhaps surprisingly,
very little is known about this problem from an approximation viewpoint.
For dimension $d=2$, we obtain an $O(k^{1/2} \cdot \log n)$-approximation
algorithm, based on a strengthening of the spreading-metric LP for 2-dimAP. The
integrality gap for this LP is shown to be $\Omega(k^{1/4})$. We also show that
it is NP-hard to approximate 2-dimAP+ within a factor better than
\Omega(k^{1/4-\eps}). We also consider a (conceptually harder, but
practically even more interesting) variant of 2-dimAP+, where the target space
is the grid $\mathbb{Z}_{\sqrt{n}} \times \mathbb{Z}_{\sqrt{n}}$, instead of
the entire integer lattice $\mathbb{Z}^2$. For this problem, we obtain a $O(k
\cdot \log^2{n})$-approximation using the same LP relaxation. We complement
this upper bound by showing an integrality gap of $\Omega(k^{1/2})$, and an
\Omega(k^{1/2-\eps})-inapproximability result.
Our results naturally extend to the case of arbitrary fixed target dimension
$d\geq 1$

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