Domination Analysis of Greedy Heuristics For The Frequency Assignment Problem

We introduce the greedy expectation algorithm for the fixed spectrum version of the frequency assignment problem. This algorithm was previously studied for the travelling salesman problem. We show that the domination number of this algorithm is at least $\sigma^{n-\lceil\log_2 n\rceil-1}$ where $\sigma$ is the available span and $n$ the number of vertices in the constraint graph. In contrast to this we show that the standard greedy algorithm has domination number strictly less than $\sigma^{n}e^{-\frac{5(n-1)}{144}}$ for large n and fixed $\sigma$

Expected Supremum of a Random Linear Combination of Shifted Kernels

We address the expected supremum of a linear combination of shifts of the sinc kernel with random coefficients. When the coefficients are Gaussian, the expected supremum is of order \sqrt{\log n}, where n is the number of shifts. When the coefficients are uniformly bounded, the expected supremum is of order \log\log n. This is a noteworthy difference to orthonormal functions on the unit interval, where the expected supremum is of order \sqrt{n\log n} for all reasonable coefficient statistics.Comment: To appear in the Journal of Fourier Analysis and Application

Optimal construction of k-nearest neighbor graphs for identifying noisy clusters

We study clustering algorithms based on neighborhood graphs on a random sample of data points. The question we ask is how such a graph should be constructed in order to obtain optimal clustering results. Which type of neighborhood graph should one choose, mutual k-nearest neighbor or symmetric k-nearest neighbor? What is the optimal parameter k? In our setting, clusters are defined as connected components of the t-level set of the underlying probability distribution. Clusters are said to be identified in the neighborhood graph if connected components in the graph correspond to the true underlying clusters. Using techniques from random geometric graph theory, we prove bounds on the probability that clusters are identified successfully, both in a noise-free and in a noisy setting. Those bounds lead to several conclusions. First, k has to be chosen surprisingly high (rather of the order n than of the order log n) to maximize the probability of cluster identification. Secondly, the major difference between the mutual and the symmetric k-nearest neighbor graph occurs when one attempts to detect the most significant cluster only.Comment: 31 pages, 2 figure

On the Capacity of the Binary-Symmetric Parallel-Relay Network

We investigate the binary-symmetric parallel-relay network where there is one source, one destination, and multiple relays in parallel. We show that forwarding relays, where the relays merely transmit their received signals, achieve the capacity in two ways: with coded transmission at the source and a finite number of relays, or uncoded transmission at the source and a sufficiently large number of relays. On the other hand, decoding relays, where the relays decode the source message, re-encode, and forward it to the destination, achieve the capacity when the number of relays is small. In addition, we show that any coding scheme that requires decoding at any relay is suboptimal in large parallel-relay networks, where forwarding relays achieve strictly higher rates.Comment: Author's final version (to appear in Transactions on Emerging Telecommunications Technologies

A Statistically Modelling Method for Performance Limits in Sensor Localization

In this paper, we study performance limits of sensor localization from a novel perspective. Specifically, we consider the Cramer-Rao Lower Bound (CRLB) in single-hop sensor localization using measurements from received signal strength (RSS), time of arrival (TOA) and bearing, respectively, but differently from the existing work, we statistically analyze the trace of the associated CRLB matrix (i.e. as a scalar metric for performance limits of sensor localization) by assuming anchor locations are random. By the Central Limit Theorems for $U$-statistics, we show that as the number of the anchors increases, this scalar metric is asymptotically normal in the RSS/bearing case, and converges to a random variable which is an affine transformation of a chi-square random variable of degree 2 in the TOA case. Moreover, we provide formulas quantitatively describing the relationship among the mean and standard deviation of the scalar metric, the number of the anchors, the parameters of communication channels, the noise statistics in measurements and the spatial distribution of the anchors. These formulas, though asymptotic in the number of the anchors, in many cases turn out to be remarkably accurate in predicting performance limits, even if the number is small. Simulations are carried out to confirm our results

Error Exponent in Asymmetric Quantum Hypothesis Testing and Its Application to Classical-Quantum Channel coding

In the simple quantum hypothesis testing problem, upper bound with asymmetric setting is shown by using a quite useful inequality by Audenaert et al, quant-ph/0610027, which was originally invented for symmetric setting. Using this upper bound, we obtain the Hoeffding bound, which are identical with the classical counter part if the hypotheses, composed of two density operators, are mutually commutative. Our upper bound improves the bound by Ogawa-Hayashi, and also provides a simpler proof of the direct part of the quantum Stein's lemma. Further, using this bound, we obtain a better exponential upper bound of the average error probability of classical-quantum channel coding

Derandomized Construction of Combinatorial Batch Codes

Combinatorial Batch Codes (CBCs), replication-based variant of Batch Codes introduced by Ishai et al. in STOC 2004, abstracts the following data distribution problem: $n$ data items are to be replicated among $m$ servers in such a way that any $k$ of the $n$ data items can be retrieved by reading at most one item from each server with the total amount of storage over $m$ servers restricted to $N$. Given parameters $m, c,$ and $k$, where $c$ and $k$ are constants, one of the challenging problems is to construct $c$-uniform CBCs (CBCs where each data item is replicated among exactly $c$ servers) which maximizes the value of $n$. In this work, we present explicit construction of $c$-uniform CBCs with $\Omega(m^{c-1+{1 \over k}})$ data items. The construction has the property that the servers are almost regular, i.e., number of data items stored in each server is in the range $[{nc \over m}-\sqrt{{n\over 2}\ln (4m)}, {nc \over m}+\sqrt{{n \over 2}\ln (4m)}]$. The construction is obtained through better analysis and derandomization of the randomized construction presented by Ishai et al. Analysis reveals almost regularity of the servers, an aspect that so far has not been addressed in the literature. The derandomization leads to explicit construction for a wide range of values of $c$ (for given $m$ and $k$) where no other explicit construction with similar parameters, i.e., with $n = \Omega(m^{c-1+{1 \over k}})$, is known. Finally, we discuss possibility of parallel derandomization of the construction

Evaluation of shared genetic susceptibility loci between autoimmune diseases and schizophrenia based on genome-wide association studies.

BACKGROUND: Epidemiological studies have documented higher than expected comorbidity (or, in some cases, inverse comorbidity) between schizophrenia and several autoimmune disorders. It remains unknown whether this comorbidity reflects shared genetic susceptibility loci. AIMS: The present study aimed to investigate whether verified genome wide significant variants of autoimmune disorders confer risk of schizophrenia, which could suggest a common genetic basis. METHODS: Seven hundred and fourteen genome wide significant risk variants of 25 autoimmune disorders were extracted from the NHGRI GWAS catalogue and examined for association to schizophrenia in the Psychiatric Genomics Consortium schizophrenia GWAS samples (36,989 cases and 113,075 controls). RESULTS: Two independent loci at 4q24 and 6p21.32-33 originally identified from GWAS of autoimmune diseases were found genome wide associated with schizophrenia (1.7 × 10(-8 )≥( )p ≥ 4.0 × 10(-21)). While these observations confirm the existence of shared genetic susceptibility loci between schizophrenia and autoimmune diseases, the findings did not show a significant enrichment. CONCLUSION: The findings do not support a genetic overlap in common SNPs between autoimmune diseases and schizophrenia that in part could explain the observed comorbidity from epidemiological studies

Sampling in a Quantum Population, and Applications

We propose a framework for analyzing classical sampling strategies for estimating the Hamming weight of a large string, when applied to a multi-qubit quantum system instead. The framework shows how to interpret such a strategy and how to define its accuracy when applied to a quantum system. Furthermore, we show how the accuracy of any strategy relates to its accuracy in its classical usage, which is well understood for the important examples. We show the usefulness of our framework by using it to obtain new and simple security proofs for the following quantum-cryptographic schemes: quantum oblivious-transfer from bit-commitment, and BB84 quantum-key-distribution