Distributed Computability in Byzantine Asynchronous Systems

In this work, we extend the topology-based approach for characterizing computability in asynchronous crash-failure distributed systems to asynchronous Byzantine systems. We give the first theorem with necessary and sufficient conditions to solve arbitrary tasks in asynchronous Byzantine systems where an adversary chooses faulty processes. In our adversarial formulation, outputs of non-faulty processes are constrained in terms of inputs of non-faulty processes only. For colorless tasks, an important subclass of distributed problems, the general result reduces to an elegant model that effectively captures the relation between the number of processes, the number of failures, as well as the topological structure of the task's simplicial complexes.Comment: Will appear at the Proceedings of the 46th Annual Symposium on the Theory of Computing, STOC 201

Constant Factor Approximation for Balanced Cut in the PIE model

We propose and study a new semi-random semi-adversarial model for Balanced Cut, a planted model with permutation-invariant random edges (PIE). Our model is much more general than planted models considered previously. Consider a set of vertices V partitioned into two clusters $L$ and $R$ of equal size. Let $G$ be an arbitrary graph on $V$ with no edges between $L$ and $R$. Let $E_{random}$ be a set of edges sampled from an arbitrary permutation-invariant distribution (a distribution that is invariant under permutation of vertices in $L$ and in $R$). Then we say that $G + E_{random}$ is a graph with permutation-invariant random edges. We present an approximation algorithm for the Balanced Cut problem that finds a balanced cut of cost $O(|E_{random}|) + n \text{polylog}(n)$ in this model. In the regime when $|E_{random}| = \Omega(n \text{polylog}(n))$, this is a constant factor approximation with respect to the cost of the planted cut.Comment: Full version of the paper at the 46th ACM Symposium on the Theory of Computing (STOC 2014). 32 page

An O(log⁡n)O(\log n)-approximation for the Set Cover Problem with Set Ownership

In highly distributed Internet measurement systems distributed agents periodically measure the Internet using a tool called {\tt traceroute}, which discovers a path in the network graph. Each agent performs many traceroute measurement to a set of destinations in the network, and thus reveals a portion of the Internet graph as it is seen from the agent locations. In every period we need to check whether previously discovered edges still exist in this period, a process termed {\em validation}. For this end we maintain a database of all the different measurements performed by each agent. Our aim is to be able to {\em validate} the existence of all previously discovered edges in the minimum possible time. In this work we formulate the validation problem as a generalization of the well know set cover problem. We reduce the set cover problem to the validation problem, thus proving that the validation problem is ${\cal NP}$-hard. We present a $O(\log n)$-approximation algorithm to the validation problem, where $n$ in the number of edges that need to be validated. We also show that unless ${\cal P = NP}$ the approximation ratio of the validation problem is $\Omega(\log n)$

Distributional Property Testing in a Quantum World

A fundamental problem in statistics and learning theory is to test properties of distributions. We show that quantum computers can solve such problems with significant speed-ups. We also introduce a novel access model for quantum distributions, enabling the coherent preparation of quantum samples, and propose a general framework that can naturally handle both classical and quantum distributions in a unified manner. Our framework generalizes and improves previous quantum algorithms for testing closeness between unknown distributions, testing independence between two distributions, and estimating the Shannon / von Neumann entropy of distributions. For classical distributions our algorithms significantly improve the precision dependence of some earlier results. We also show that in our framework procedures for classical distributions can be directly lifted to the more general case of quantum distributions, and thus obtain the first speed-ups for testing properties of density operators that can be accessed coherently rather than only via sampling

Approximate Near Neighbors for General Symmetric Norms

We show that every symmetric normed space admits an efficient nearest neighbor search data structure with doubly-logarithmic approximation. Specifically, for every $n$, $d = n^{o(1)}$, and every $d$-dimensional symmetric norm $\|\cdot\|$, there exists a data structure for $\mathrm{poly}(\log \log n)$-approximate nearest neighbor search over $\|\cdot\|$ for $n$-point datasets achieving $n^{o(1)}$ query time and $n^{1+o(1)}$ space. The main technical ingredient of the algorithm is a low-distortion embedding of a symmetric norm into a low-dimensional iterated product of top-$k$ norms. We also show that our techniques cannot be extended to general norms.Comment: 27 pages, 1 figur

Quantum Discord and Quantum Computing - An Appraisal

We discuss models of computing that are beyond classical. The primary motivation is to unearth the cause of nonclassical advantages in computation. Completeness results from computational complexity theory lead to the identification of very disparate problems, and offer a kaleidoscopic view into the realm of quantum enhancements in computation. Emphasis is placed on the `power of one qubit' model, and the boundary between quantum and classical correlations as delineated by quantum discord. A recent result by Eastin on the role of this boundary in the efficient classical simulation of quantum computation is discussed. Perceived drawbacks in the interpretation of quantum discord as a relevant certificate of quantum enhancements are addressed.Comment: To be published in the Special Issue of the International Journal of Quantum Information on "Quantum Correlations: entanglement and beyond." 11 pages, 4 figure