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 $T$ of the given network $G$ such that subsequent to the failure of a single edge or vertex, the surviving part $T'$ of $T$ still contains a BFS spanning tree for (the surviving part of) $G$. Our main results are as follows. We present an algorithm that for every $n$-vertex graph $G$ and source node $s$ constructs a (single edge failure) FT-BFS tree rooted at $s$ with O(n \cdot \min\{\Depth(s), \sqrt{n}\}) edges, where \Depth(s) is the depth of the BFS tree rooted at $s$. This result is complemented by a matching lower bound, showing that there exist $n$-vertex graphs with a source node $s$ for which any edge (or vertex) FT-BFS tree rooted at $s$ has $\Omega(n^{3/2})$ 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 $s\in S$ for some subset of sources $S\subseteq V$. Again, tight bounds are provided, showing that there exists a poly-time algorithm that for every $n$-vertex graph and source set $S \subseteq V$ of size $\sigma$ constructs a (single failure) FT-MBFS tree $T^*(S)$ from each source $s_i \in S$, with $O(\sqrt{\sigma} \cdot n^{3/2})$ edges, and on the other hand there exist $n$-vertex graphs with source sets $S \subseteq V$ of cardinality $\sigma$, on which any FT-MBFS tree from $S$ has $\Omega(\sqrt{\sigma}\cdot n^{3/2})$ edges. Finally, we propose an $O(\log n)$ approximation algorithm for constructing FT-BFS and FT-MBFS structures. The latter is complemented by a hardness result stating that there exists no $\Omega(\log n)$ approximation algorithm for these problems under standard complexity assumptions

Stable scalable control of soliton propagation in broadband nonlinear optical waveguides

We develop a method for achieving scalable transmission stabilization and switching of $N$ colliding soliton sequences in optical waveguides with broadband delayed Raman response and narrowband nonlinear gain-loss. We show that dynamics of soliton amplitudes in $N$-sequence transmission is described by a generalized $N$-dimensional predator-prey model. Stability and bifurcation analysis for the predator-prey model are used to obtain simple conditions on the physical parameters for robust transmission stabilization as well as on-off and off-on switching of $M$ out of $N$ soliton sequences. Numerical simulations for single-waveguide transmission with a system of $N$ coupled nonlinear Schr\"odinger equations with $2 \le N \le 4$ show excellent agreement with the predator-prey model's predictions and stable propagation over significantly larger distances compared with other broadband nonlinear single-waveguide systems. Moreover, stable on-off and off-on switching of multiple soliton sequences and stable multiple transmission switching events are demonstrated by the simulations. We discuss the reasons for the robustness and scalability of transmission stabilization and switching in waveguides with broadband delayed Raman response and narrowband nonlinear gain-loss, and explain their advantages compared with other broadband nonlinear waveguides.Comment: 37 pages, 7 figures, Eur. Phys. J. D (accepted

A Mean-field Approach for an Intercarrier Interference Canceller for OFDM

The similarity of the mathematical description of random-field spin systems to orthogonal frequency-division multiplexing (OFDM) scheme for wireless communication is exploited in an intercarrier-interference (ICI) canceller used in the demodulation of OFDM. The translational symmetry in the Fourier domain generically concentrates the major contribution of ICI from each subcarrier in the subcarrier's neighborhood. This observation in conjunction with mean field approach leads to a development of an ICI canceller whose necessary cost of computation scales linearly with respect to the number of subcarriers. It is also shown that the dynamics of the mean-field canceller are well captured by a discrete map of a single macroscopic variable, without taking the spatial and time correlations of estimated variables into account.Comment: 7pages, 3figure

Parallel Metric Tree Embedding based on an Algebraic View on Moore-Bellman-Ford

A \emph{metric tree embedding} of expected \emph{stretch~$\alpha \geq 1$} maps a weighted $n$-node graph $G = (V, E, \omega)$ to a weighted tree $T = (V_T, E_T, \omega_T)$ with $V \subseteq V_T$ such that, for all $v,w \in V$, $\operatorname{dist}(v, w, G) \leq \operatorname{dist}(v, w, T)$ and $operatorname{E}[\operatorname{dist}(v, w, T)] \leq \alpha \operatorname{dist}(v, w, G)$. Such embeddings are highly useful for designing fast approximation algorithms, as many hard problems are easy to solve on tree instances. However, to date the best parallel $(\operatorname{polylog} n)$-depth algorithm that achieves an asymptotically optimal expected stretch of $\alpha \in \operatorname{O}(\log n)$ requires $\operatorname{\Omega}(n^2)$ work and a metric as input. In this paper, we show how to achieve the same guarantees using $\operatorname{polylog} n$ depth and $\operatorname{\tilde{O}}(m^{1+\epsilon})$ work, where $m = |E|$ and $\epsilon > 0$ is an arbitrarily small constant. Moreover, one may further reduce the work to $\operatorname{\tilde{O}}(m + n^{1+\epsilon})$ at the expense of increasing the expected stretch to $\operatorname{O}(\epsilon^{-1} \log n)$. Our main tool in deriving these parallel algorithms is an algebraic characterization of a generalization of the classic Moore-Bellman-Ford algorithm. We consider this framework, which subsumes a variety of previous "Moore-Bellman-Ford-like" algorithms, to be of independent interest and discuss it in depth. In our tree embedding algorithm, we leverage it for providing efficient query access to an approximate metric that allows sampling the tree using $\operatorname{polylog} n$ depth and $\operatorname{\tilde{O}}(m)$ work. We illustrate the generality and versatility of our techniques by various examples and a number of additional results

Simulations of the Angular Dependence of the Dipole-Dipole Interaction

In our project we ran computations on a supercomputer to simulate experiments performed on highly excited atoms at μK temperatures. At μK temperatures the atoms are moving slowly so there are essentially no collisions of the atoms on the time scales at which we perform our experiments. In the absence of collisions the atoms exchange energy through long range dipole-dipole interactions. This exchange depends on the distances between and relative orientation of the atoms. The angular dependence between two atoms has recently been studied experimentally1 . We simulate experimentally accessible spatial arrangements to see if the effect of the angular dependence can be measured in the many atom case. We present results that show that the angular dependence has a measurable effect on the time evolution of the spatial distribution of the energy in the system. 1. arXiv:1504.00301[physics.atom-ph

Simulations of the Angular Dependence of the Dipole-Dipole Interaction

In our project we ran computations on a supercomputer to simulate experiments performed on highly excited atoms at μK temperatures. At μK temperatures the atoms are moving slowly so there are essentially no collisions of the atoms on the time scales at which we perform our experiments. In the absence of collisions the atoms exchange energy through long range dipole-dipole interactions. This exchange depends on the distances between and relative orientation of the atoms. The angular dependence between two atoms has recently been studied experimentally1 . We simulate experimentally accessible spatial arrangements to see if the effect of the angular dependence can be measured in the many atom case. We present results that show that the angular dependence has a measurable effect on the time evolution of the spatial distribution of the energy in the system. 1. arXiv:1504.00301[physics.atom-ph

Simulations of the Angular Dependence of the Dipole-Dipole Interaction

We conducted simulations of Rydberg atoms in a magneto-optical trap using the supercomputer available on campus and the COMET supercomputer provided by the NSF. Our research focused on the angular dependence of the long range interaction between Rydberg atoms. We simulated randomly distributed atoms alligned with a magnetic and electric field. We compared the simulated interaction rates for different electric field directions

Fast two-pulse collisions in linear diffusion-advection systems with weak quadratic loss in spatial dimension 2

We investigate the dynamics of fast two-pulse collisions in linear diffusion-advection systems with weak quadratic loss in spatial dimension 2. We introduce a two-dimensional perturbation method, which generalizes the perturbation method used for studying two-pulse collisions in spatial dimension 1. We then use the generalized perturbation method to show that a fast collision in spatial dimension 2 leads to a change in the pulse shape in the direction transverse to the advection velocity vector. Moreover, we show that in the important case of a separable initial condition, the longitudinal part in the expression for the amplitude shift has a simple universal form, while the transverse part does not. Additionally, we show that anisotropy in the initial condition leads to a complex dependence of the amplitude shift on the orientation angle between the pulses. Our perturbation theory predictions are in very good agreement with results of extensive numerical simulations with the weakly perturbed diffusion-advection model. Thus, our study significantly enhances and generalizes the results of previous works on fast collisions in diffusion-advection systems, which were limited to spatial dimension 1.Comment: The paper presents a perturbation method for fast two-pulse collisions in diffusion-advection systems with weak nonlinear loss in dimension higher than 1. It generalizes the method that was used in arXiv:1702.05583 and arXiv:1808.04323 for the 1-dimensional problem. It complements the method that was presented in arXiv:2102.07438 for the high-dimensional problem in weakly nonlinear optical medi

Prefect Klein tunneling in anisotropic graphene-like photonic lattices

We study the scattering of waves off a potential step in deformed honeycomb lattices. For small deformations below a critical value, perfect Klein tunneling is obtained. This means that a potential step in any direction transmits waves at normal incidence with unit transmission probability, irrespective of the details of the potential. Beyond the critical deformation, a gap in the spectrum is formed, and a potential step in the deformation direction reflects all normal-incidence waves, exhibiting a dramatic transition form unit transmission to total reflection. These phenomena are generic to honeycomb lattice systems, and apply to electromagnetic waves in photonic lattices, quasi-particles in graphene, cold atoms in optical lattices