Tight Lower Bounds for Data-Dependent Locality-Sensitive Hashing

We prove a tight lower bound for the exponent $\rho$ for data-dependent Locality-Sensitive Hashing schemes, recently used to design efficient solutions for the $c$-approximate nearest neighbor search. In particular, our lower bound matches the bound of $\rho\le \frac{1}{2c-1}+o(1)$ for the $\ell_1$ space, obtained via the recent algorithm from [Andoni-Razenshteyn, STOC'15]. In recent years it emerged that data-dependent hashing is strictly superior to the classical Locality-Sensitive Hashing, when the hash function is data-independent. In the latter setting, the best exponent has been already known: for the $\ell_1$ space, the tight bound is $\rho=1/c$, with the upper bound from [Indyk-Motwani, STOC'98] and the matching lower bound from [O'Donnell-Wu-Zhou, ITCS'11]. We prove that, even if the hashing is data-dependent, it must hold that $\rho\ge \frac{1}{2c-1}-o(1)$. To prove the result, we need to formalize the exact notion of data-dependent hashing that also captures the complexity of the hash functions (in addition to their collision properties). Without restricting such complexity, we would allow for obviously infeasible solutions such as the Voronoi diagram of a dataset. To preclude such solutions, we require our hash functions to be succinct. This condition is satisfied by all the known algorithmic results.Comment: 16 pages, no figure

Hydrodynamic spectrum of a superfluid in an elongated trap

--In this article we study the hydrodynamic spectrum of a superfluid confined in a cylindrical trap. We show that the dispersion relation $\omega$(q) of the phonon branch scales like \sqrt q at large q, leading to a vanishingly small superfluid critical velocity. In practice the critical velocity is set by the breakdown of the hydrodynamic approximation. For a broad class of superfluids, this entails a reduction of the critical velocity by a factor ($\omega$ $\perp$ /\"i1/2c) 1/3 with respect to the free-space prediction (here $\omega$ $\perp$ is the trapping frequency and \"i1/2c the chemical potential of the cloud)

Seesaw Mechanism in Warped Geometry

We show how the seesaw mechanism for neutrino masses can be realized within a five dimensional (5D) warped geometry framework. Intermediate scale standard model (SM) singlet neutrino masses, needed to explain the atmospheric and solar neutrino oscillations, are shown to be proportional to M_Pl\exp((2c-1)\pi kR), where c denotes the coefficient of the 5D Dirac mass term for the singlet neutrino which also has a Planck scale Majorana mass localized on the Planck-brane, and kR~11 in order to resolve the gauge hierarchy problem. The case with a bulk 5D Majorana mass term for the singlet neutrino is briefly discussed.Comment: 14 pages, LaTeX, 4 figures, references adde

Perfect graphs of fixed density: counting and homogenous sets

For c in [0,1] let P_n(c) denote the set of n-vertex perfect graphs with density c and C_n(c) the set of n-vertex graphs without induced C_5 and with density c. We show that log|P_n(c)|/binom{n}{2}=log|C_n(c)|/binom{n}{2}=h(c)+o(1) with h(c)=1/2 if 1/4<c<3/4 and h(c)=H(|2c-1|)/2 otherwise, where H is the binary entropy function. Further, we use this result to deduce that almost all graphs in C_n(c) have homogenous sets of linear size. This answers a question raised by Loebl, Reed, Scott, Thomason, and Thomass\'e [Almost all H-free graphs have the Erd\H{o}s-Hajnal property] in the case of forbidden induced C_5.Comment: 19 page

Optimal Data-Dependent Hashing for Approximate Near Neighbors

We show an optimal data-dependent hashing scheme for the approximate near neighbor problem. For an $n$-point data set in a $d$-dimensional space our data structure achieves query time $O(d n^{\rho+o(1)})$ and space $O(n^{1+\rho+o(1)} + dn)$, where $\rho=\tfrac{1}{2c^2-1}$ for the Euclidean space and approximation $c>1$. For the Hamming space, we obtain an exponent of $\rho=\tfrac{1}{2c-1}$. Our result completes the direction set forth in [AINR14] who gave a proof-of-concept that data-dependent hashing can outperform classical Locality Sensitive Hashing (LSH). In contrast to [AINR14], the new bound is not only optimal, but in fact improves over the best (optimal) LSH data structures [IM98,AI06] for all approximation factors $c>1$. From the technical perspective, we proceed by decomposing an arbitrary dataset into several subsets that are, in a certain sense, pseudo-random.Comment: 36 pages, 5 figures, an extended abstract appeared in the proceedings of the 47th ACM Symposium on Theory of Computing (STOC 2015

Characterization of crested wheatgrass germplasms for plant maturity and associated physiological and morphological traits

Crested wheatgrass [Agropyron cristatum (L.) Gaertn.] is a drought tolerant, winter hardy perennial grass used for early spring grazing in western Canada. This grass matures early, and mature plants are not palatable for grazing animals. The objectives of this study were: 1) determine DNA content and ploidy level of 45 crested wheatgrass accessions 2) to characterize crested wheatgrass germplasm for plant maturity and associated agronomic characteristics to identify superior germplasm with late maturity; 3) to evaluate flowering time of selected germplasms of crested wheatgrass under a controlled environment. A field plot was established using 45 crested wheatgrass accessions in July 2014 at Agriculture and Agri-Food Canada (AAFC) Saskatoon Research Center at Saskatoon SK, Canada using a randomized complete block design with four replications with data collected in 2015, 2016 and 2017. On the basis of DNA content (pg 2C-1 =DNA content of diploid somatic nucleus), mean DNA content was 14.12 pg 2C-1 for diploid, 28.02 pg 2C-1 and 39.48 pg 2C-1 for tetraploid and hexaploid crested wheatgrass, respectively. Among the 45 accessions, there were 8 diploid, 31 tetraploid, and 6 hexaploid accessions. Plant maturity and other measured characteristics differed significantly among the ploidy levels. Days to heading, plant height, leaf-to-stem ratio, forage DM yield, leafiness and plant vigor and nutritive value (crude protein, neutral detergent and acid detergent fibers) differed significantly (P ≤ 0.05) among accessions at flowering stage. In this study, days to heading showed a positive correlation with leaf-to-stem ratio (r=0.23, P<0.0001), indicating that selection for later maturity in crested wheatgrass may lead to an increase in leafiness. When all 45 accessions were considered, there was a non-significant correlation between days to heading and DM yield (r= 0.07, P=0.09), but this relationship was significant (r=0.34, P<0.0001) when only Canadian breeding lines and cultivars were considered. Based on agronomic performance and nutritive value, the 45 crested wheatgrass accessions were grouped into three main clusters. In addition, ranking of days to heading among selected accessions was consistent in field and controlled environments. In conclusion, plant maturity varied within- and among- accessions, among ploidy levels, and selection for late maturity may simultaneously increase forage DM yield and leaf-to-stem ratio in crested wheatgrass. Information obtained from this study on agro-morphological traits, nutritive values and ploidy determination among the 45 crested wheatgrass accessions will be useful for future crested wheatgrass breeding programs

Constant-factor approximations of branch-decomposition and largest grid minor of planar graphs in O(n1+ϵ) time

AbstractWe give constant-factor approximation algorithms for computing the optimal branch-decompositions and largest grid minors of planar graphs. For a planar graph G with n vertices, let bw(G) be the branchwidth of G and gm(G) the largest integer g such that G has a g×g grid as a minor. Let c≥1 be a fixed integer and α,β arbitrary constants satisfying α>c+1 and β>2c+1. We give an algorithm which constructs in O(n1+1clogn) time a branch-decomposition of G with width at most αbw(G). We also give an algorithm which constructs a g×g grid minor of G with g≥gm(G)β in O(n1+1clogn) time. The constants hidden in the Big-O notations are proportional to cα−(c+1) and cβ−(2c+1), respectively