Spanners and Sparsifiers in Dynamic Streams

Linear sketching is a popular technique for computing in dynamic streams, where one needs to handle both insertions and deletions of elements. The underlying idea of taking randomized linear measurements of input data has been extremely successful in providing space-efficient algorithms for classical problems such as frequency moment estimation and computing heavy hitters, and was very recently shown to be a powerful technique for solving graph problems in dynamic streams [AGM’12]. Ideally, one would like to obtain algorithms that use one or a small constant number of passes over the data and a small amount of space (i.e. sketching dimension) to preserve some useful properties of the input graph presented as a sequence of edge insertions and edge deletions. In this paper, we concentrate on the problem of constructing linear sketches of graphs that (approximately) preserve the spectral information of the graph in a few passes over the stream. We do so by giving the first sketch-based algorithm for constructing multiplicative graph spanners in only two passes over the stream. Our spanners use Õ(n1+1/k) bits of space and have stretch 2 k. While this stretch is larger than the conjectured optimal 2k − 1 for this amount of space, we show for an appropriate k that it implies the first 2-pass spectral sparsifier with n 1+o(1) bits of space. Previous constructions of spectral sparsifiers in this model with a constant number of passes would require n 1+c bits of space for a constant c> 0. We also give an algorithm for constructing spanners that provides an additive approximation to the shortest path metric using a single pass over the data stream, also achieving an essentially best possible space/approximation tradeoff. 1

Optimal lower bounds for universal relation, and for samplers and finding duplicates in streams

In the communication problem $\mathbf{UR}$ (universal relation) [KRW95], Alice and Bob respectively receive $x, y \in\{0,1\}^n$ with the promise that $x\neq y$. The last player to receive a message must output an index $i$ such that $x_i\neq y_i$. We prove that the randomized one-way communication complexity of this problem in the public coin model is exactly $\Theta(\min\{n,\log(1/\delta)\log^2(\frac n{\log(1/\delta)})\})$ for failure probability $\delta$. Our lower bound holds even if promised $\mathop{support}(y)\subset \mathop{support}(x)$. As a corollary, we obtain optimal lower bounds for $\ell_p$-sampling in strict turnstile streams for $0\le p < 2$, as well as for the problem of finding duplicates in a stream. Our lower bounds do not need to use large weights, and hold even if promised $x\in\{0,1\}^n$ at all points in the stream. We give two different proofs of our main result. The first proof demonstrates that any algorithm $\mathcal A$ solving sampling problems in turnstile streams in low memory can be used to encode subsets of $[n]$ of certain sizes into a number of bits below the information theoretic minimum. Our encoder makes adaptive queries to $\mathcal A$ throughout its execution, but done carefully so as to not violate correctness. This is accomplished by injecting random noise into the encoder's interactions with $\mathcal A$, which is loosely motivated by techniques in differential privacy. Our second proof is via a novel randomized reduction from Augmented Indexing [MNSW98] which needs to interact with $\mathcal A$ adaptively. To handle the adaptivity we identify certain likely interaction patterns and union bound over them to guarantee correct interaction on all of them. To guarantee correctness, it is important that the interaction hides some of its randomness from $\mathcal A$ in the reduction.Comment: merge of arXiv:1703.08139 and of work of Kapralov, Woodruff, and Yahyazade

Noisy Boolean Hidden Matching with Applications

The Boolean Hidden Matching (BHM) problem, introduced in a seminal paper of Gavinsky et al. [STOC\u2707], has played an important role in lower bounds for graph problems in the streaming model (e.g., subgraph counting, maximum matching, MAX-CUT, Schatten p-norm approximation). The BHM problem typically leads to ?(?n) space lower bounds for constant factor approximations, with the reductions generating graphs that consist of connected components of constant size. The related Boolean Hidden Hypermatching (BHH) problem provides ?(n^{1-1/t}) lower bounds for 1+O(1/t) approximation, for integers t ? 2. The corresponding reductions produce graphs with connected components of diameter about t, and essentially show that long range exploration is hard in the streaming model with an adversarial order of updates. In this paper we introduce a natural variant of the BHM problem, called noisy BHM (and its natural noisy BHH variant), that we use to obtain stronger than ?(?n) lower bounds for approximating a number of the aforementioned problems in graph streams when the input graphs consist only of components of diameter bounded by a fixed constant. We next introduce and study the graph classification problem, where the task is to test whether the input graph is isomorphic to a given graph. As a first step, we use the noisy BHM problem to show that the problem of classifying whether an underlying graph is isomorphic to a complete binary tree in insertion-only streams requires ?(n) space, which seems challenging to show using either BHM or BHH

Asymmetric Faraday Effect in a Magnetophotonic Crystal

It is widely known that the magneto-optical Faraday effect is linear in magnetization and therefore the Faraday angles for the states with opposite magnetizations are of opposite sign but equal in modulus. Here we experimentally study propagation of light through a one-dimensional all-garnet magnetophotonic crystal to demonstrate an asymmetric Faraday effect (AFE) for which Faraday angles for opposite magnetic states differ not only in sign but in the absolute value as well. AFE appears in the vicinity of the cavity resonance for an oblique incidence of light which plane of polarization is inclined to the incidence plane. Under proper incidence and polarization angles the magnitude of AFE could be very large reaching 30% of the absolute value of the Faraday effect. The effect originates from the difference in Q-factors for p- and s- polarized cavity modes that breaks the symmetry between the two opposite directions of polarization rotation. The discovered AFE is of prime importance for nanoscale magnonics and optomagnetism.Comment: Supplementary information provided after the main tex

Carbon nanotubes for stabilization of nanostructured lipid particles

Carbon nanotubes (CNTs) are increasingly studied for innovative biotechnological applications particularly where they are combined with essential biological materials like lipids. Lipids have been used earlier for enhancing the dispersibility of CNTs in aqueous solutions. Here we report a novel application of CNTs for stabilization of internally self-assembled nanostructured lipid particles of 2–5 μm size. Single-walled (pristine) as well as –OH and –COOH functionalized multi-walled CNTs were employed to produce nanostructured emulsions which stayed stable for months and could be re-dispersed after complete dehydration. Concentrations of CNTs employed for stabilization were very low; moreover CNTs were well-decorated with lipid molecules. These features contribute towards reducing their toxicity and improving biocompatibility for biomedical and pharmaceutical applications. Our approach paves the way for future development of combination therapies employing both CNTs and nanostructured lipid self-assembly together as carriers of different drugs

Rubisco evolution in C₄ eudicots: an analysis of Amaranthaceae sensu lato.

BACKGROUND: Rubisco (ribulose-1,5-bisphosphate carboxylase/oxygenase) catalyses the key reaction in the photosynthetic assimilation of CO₂. In C₄ plants CO₂ is supplied to Rubisco by an auxiliary CO₂-concentrating pathway that helps to maximize the carboxylase activity of the enzyme while suppressing its oxygenase activity. As a consequence, C₄ Rubisco exhibits a higher maximum velocity but lower substrate specificity compared with the C₃ enzyme. Specific amino-acids in Rubisco are associated with C₄ photosynthesis in monocots, but it is not known whether selection has acted on Rubisco in a similar way in eudicots. METHODOLOGY/PRINCIPAL FINDINGS: We investigated Rubisco evolution in Amaranthaceae sensu lato (including Chenopodiaceae), the third-largest family of C₄ plants, using phylogeny-based maximum likelihood and Bayesian methods to detect Darwinian selection on the chloroplast rbcL gene in a sample of 179 species. Two Rubisco residues, 281 and 309, were found to be under positive selection in C₄ Amaranthaceae with multiple parallel replacements of alanine by serine at position 281 and methionine by isoleucine at position 309. Remarkably, both amino-acids have been detected in other C₄ plant groups, such as C₄ monocots, illustrating a striking parallelism in molecular evolution. CONCLUSIONS/SIGNIFICANCE: Our findings illustrate how simple genetic changes can contribute to the evolution of photosynthesis and strengthen the hypothesis that parallel amino-acid replacements are associated with adaptive changes in Rubisco

Cyclotron resonance overtones and near-field magnetoabsorption via terahertz Bernstein modes in graphene

Two-dimensional electron systems subjected to a perpendicular magnetic field absorb electromagnetic radiation via the cyclotron resonance (CR). Here we report a qualitative breach of this well-known behaviour in graphene. Our study of the terahertz photoresponse reveals a resonant burst at the main overtone of the CR, drastically exceeding the signal detected at the position of the ordinary CR. In accordance with the developed theory, the photoresponse dependencies on the magnetic field, doping level, and sample geometry suggest that the origin of this anomaly lies in the near-field magnetoabsorption facilitated by the Bernstein modes, ultra-slow magnetoplasmonic excitations reshaped by nonlocal electron dynamics. Close to the CR harmonics, these modes are characterized by a flat dispersion and a diverging plasmonic density of states that strongly amplifies the radiation absorption. Besides fundamental interest, our experimental results and developed theory show that the radiation absorption via nonlocal collective modes can facilitate a strong photoresponse, a behaviour potentially useful for infrared and terahertz technology.Comment: 27 pages, 22 figure