OpenRaiser/Intern-Atlas huggingface dataset:
The Stretch Factor of the Delaunay Triangulation Is Less Than 1.998
Venue: arXiv cs.CG 2013
Year: 2013

Let $S$ be a finite set of points in the Euclidean plane. Let $D$ be a Delaunay triangulation of $S$. The {\em stretch factor} (also known as {\em dilation} or {\em spanning ratio}) of $D$ is the maximum ratio, among all points $p$ and $q$ in $S$, of the shortest path distance from $p$ to $q$ in $D$ over the Euclidean distance $||pq||$. Proving a tight bound on the stretch factor of the Delaunay triangulation has been a long standing open problem in computational geometry. In this paper we prove that the stretch factor of the Delaunay triangulation of a set of points in the plane is less than $ρ= 1.998$, improving the previous best upper bound of 2.42 by Keil and Gutwin (1989). Our bound 1.998 is better than the current upper bound of 2.33 for the special case when the point set is in convex position by Cui, Kanj and Xia (2009). This upper bound breaks the barrier 2, which is significant because previously no family of plane graphs was known to have a stretch factor guaranteed to be less than 2 on any set of points.
On Sets of Lines Not-Supporting Trees
Venue: arXiv cs.DM 2013
Year: 2013

We study the following problem introduced by Dujmovic et al. Given a tree $T = (V,E)$, on $n$ vertices, a set of $n$ lines $\mathcal{L}$ in the plane and a bijection $ι: V \rightarrow \mathcal{L}$, we are asked to find a crossing-free straight-line embedding of $T$ so that $v\in ι(v)$, for all $v\in V$. We say that a set of $n$ lines $\mathcal{L}$ is universal for trees if for any tree $T$ and any bijection $ι$ there exists such an embedding. We prove that any sufficiently big set of lines is not universal for trees, which solves an open problem asked by Dujmovic et al.
Root Refinement for Real Polynomials
Venue: arXiv cs.SC 2013
Year: 2013

We consider the problem of approximating all real roots of a square-free polynomial $f$. Given isolating intervals, our algorithm refines each of them to a width of $2^{-L}$ or less, that is, each of the roots is approximated to $L$ bits after the binary point. Our method provides a certified answer for arbitrary real polynomials, only considering finite approximations of the polynomial coefficients and choosing a suitable working precision adaptively. In this way, we get a correct algorithm that is simple to implement and practically efficient. Our algorithm uses the quadratic interval refinement method; we adapt that method to be able to cope with inaccuracies when evaluating $f$, without sacrificing its quadratic convergence behavior. We prove a bound on the bit complexity of our algorithm in terms of the degree of the polynomial, the size and the separation of the roots, that is, parameters exclusively related to the geometric location of the roots. Our bound is near optimal and significantly improves previous work on integer polynomials. Furthermore, it essentially matches the best known theoretical bounds on root approximation which are obtained by very sophisticated algorithms. We also investigate the practical behavior of the algorithm and demonstrate how closely the practical performance matches our asymptotic bounds.
Adaptive Evolutionary Clustering
Venue: arXiv cs.LG 2013
Year: 2013

In many practical applications of clustering, the objects to be clustered evolve over time, and a clustering result is desired at each time step. In such applications, evolutionary clustering typically outperforms traditional static clustering by producing clustering results that reflect long-term trends while being robust to short-term variations. Several evolutionary clustering algorithms have recently been proposed, often by adding a temporal smoothness penalty to the cost function of a static clustering method. In this paper, we introduce a different approach to evolutionary clustering by accurately tracking the time-varying proximities between objects followed by static clustering. We present an evolutionary clustering framework that adaptively estimates the optimal smoothing parameter using shrinkage estimation, a statistical approach that improves a naive estimate using additional information. The proposed framework can be used to extend a variety of static clustering algorithms, including hierarchical, k-means, and spectral clustering, into evolutionary clustering algorithms. Experiments on synthetic and real data sets indicate that the proposed framework outperforms static clustering and existing evolutionary clustering algorithms in many scenarios.
Correlation Testing for Affine Invariant Properties on $\mathbb{F}_p^n$ in the High Error Regime
Venue: arXiv cs.CC 2013
Year: 2013

Recently there has been much interest in Gowers uniformity norms from the perspective of theoretical computer science. This is mainly due to the fact that these norms provide a method for testing whether the maximum correlation of a function $f:\mathbb{F}_p^n \rightarrow \mathbb{F}_p$ with polynomials of degree at most $d \le p$ is non-negligible, while making only a constant number of queries to the function. This is an instance of {\em correlation testing}. In this framework, a fixed test is applied to a function, and the acceptance probability of the test is dependent on the correlation of the function from the property. This is an analog of {\em proximity oblivious testing}, a notion coined by Goldreich and Ron, in the high error regime. In this work, we study general properties which are affine invariant and which are correlation testable using a constant number of queries. We show that any such property (as long as the field size is not too small) can in fact be tested by Gowers uniformity tests, and hence having correlation with the property is equivalent to having correlation with degree $d$ polynomials for some fixed $d$. We stress that our result holds also for non-linear properties which are affine invariant. This completely classifies affine invariant properties which are correlation testable. The proof is based on higher-order Fourier analysis. Another ingredient is a nontrivial extension of a graph theoretical theorem of Erdös, Lovász and Spencer to the context of additive number theory.
An automatic volume control for preserving intelligibility
Venue: arXiv cs.SD 2013
Year: 2013

A new method has been developed to adjust volume automatically on all audio devices equipped with at least one microphone, including mobile phones, personal media players, headsets, and car radios, that might be used in noisy environments, such as crowds, cars, and outdoors. The method uses a patented set of algorithms, implemented on the chips in such devices, to preserve constant intelligibility of speech in noisy environments, rather than constant signal-to-noise ratio. The algorithms analyze the noise background in real time and compensate only for fluctuating noise in the frequency domain and the time domain that interferes with intelligibility of speech. Advantages of this method of controlling volume include: Controlling volume without sacrificing clarity; adjusting only for persistent speech-interference noise; smoothing volume fluctuations; and eliminating static-like bursts caused by noise spikes. Practical human-factors approaches to implementing these algorithms in mobile phones are discussed.
Satisfiability of Acyclic and Almost Acyclic CNF Formulas
Venue: arXiv cs.DS 2013
Year: 2013

We show that the Satisfiability (SAT) problem for CNF formulas with β-acyclic hypergraphs can be solved in polynomial time by using a special type of Davis-Putnam resolution in which each resolvent is a subset of a parent clause. We extend this class to CNF formulas for which this type of Davis-Putnam resolution still applies and show that testing membership in this class is NP-complete. We compare the class of β-acyclic formulas and this superclass with a number of known polynomial formula classes. We then study the parameterized complexity of SAT for "almost" β-acyclic instances, using as parameter the formula's distance from being β-acyclic. As distance we use the size of a smallest strong backdoor set and the β-hypertree width. As a by-product we obtain the W[1]-hardness of SAT parameterized by the (undirected) clique-width of the incidence graph, which disproves a conjecture by Fischer, Makowsky, and Ravve.
A mixed formulation for a modification to Darcy equation based on Picard linearization and numerical solutions to large-scale realistic problems
Venue: arXiv cs.NA 2013
Year: 2013

In this paper we consider a modification to Darcy equation by taking into account the dependence of viscosity on the pressure. We present a stabilized mixed formulation for the resulting governing equations. Equal-order interpolation for the velocity and pressure is considered, and shown to be stable (which is not the case under the classical mixed formulation). The proposed mixed formulation is tested using a wide variety of numerical examples. The proposed formulation is also implemented in a parallel setting, and the performance of the formulation for large-scale problems is illustrated using a representative problem. Two practical and technologically important problems, one each on enhanced oil recovery and geological carbon-dioxide sequestration, are solved using the proposed formulation. The numerical examples show that the predictions based on Darcy model are qualitatively and quantitatively different from that of the predictions based on the modified Darcy model, which takes into account the dependence of the viscosity on the pressure. In particular, the numerical example on the geological carbon-dioxide sequestration shows that Darcy model over-predicts the leakage into an abandoned well when compared to that of the modified Darcy model. On the other hand, the modified Darcy model predicts higher pressures and higher pressure gradients near the injection well. These predictions have dire consequences in predicting damage and fracture zones, and designing the seal, whose integrity is crucial to the safety of a geological carbon-dioxide sequestration geosystem.
An Achievable Rate Region for the Broadcast Channel with Feedback
Venue: arXiv cs.IT 2013
Year: 2013

A single-letter achievable rate region is proposed for the two-receiver discrete memoryless broadcast channel with generalized feedback. The coding strategy involves block-Markov superposition coding, using Marton's coding scheme for the broadcast channel without feedback as the starting point. If the message rates in the Marton scheme are too high to be decoded at the end of a block, each receiver is left with a list of messages compatible with its output. Resolution information is sent in the following block to enable each receiver to resolve its list. The key observation is that the resolution information of the first receiver is correlated with that of the second. This correlated information is efficiently transmitted via joint source-channel coding, using ideas similar to the Han-Costa coding scheme. Using the result, we obtain an achievable rate region for the stochastically degraded AWGN broadcast channel with noisy feedback from only one receiver. It is shown that this region is strictly larger than the no-feedback capacity region.
Extended formulations, non-negative factorizations and randomized communication protocols
Venue: arXiv cs.DM 2013
Year: 2013

An extended formulation of a polyhedron $P$ is a linear description of a polyhedron $Q$ together with a linear map $π$ such that $π(Q)=P$. These objects are of fundamental importance in polyhedral combinatorics and optimization theory, and the subject of a number of studies. Yannakakis' factorization theorem [M. Yannakakis. Expressing combinatorial optimization problems by linear programs. {\em J. Comput. System Sci.}, 43(3):441--466 (1991)] provides a surprising connection between extended formulations and communication complexity, showing that the smallest size of an extended formulation of $P$ equals the nonnegative rank of its slack matrix $S$. Moreover, Yannakakis also shows that the nonnegative rank of $S$ is at most $2^c$, where $c$ is the complexity of any \emph{deterministic} protocol computing $S$. In this paper, we show that the latter result can be strengthened when we allow protocols to be \emph{randomized}. In particular, we prove that the base-2 logarithm of the nonnegative rank of any nonnegative matrix equals the minimum complexity of a randomized communication protocol computing the matrix in expectation. Using Yannakakis' factorization theorem, this implies that the base-2 logarithm of the smallest size of an extended formulation of a polytope $P$ equals the minimum complexity of a randomized communication protocol computing the slack matrix of $P$ in expectation. We show that allowing randomization in the protocol can be crucial for obtaining small extended formulations. Specifically, we prove that for the spanning tree and perfect matching polytopes, small variance in the protocol forces large size in the extended formulation.
MIMO Broadcasting for Simultaneous Wireless Information and Power Transfer
Venue: arXiv cs.IT 2013
Year: 2013

Wireless power transfer (WPT) is a promising new solution to provide convenient and perpetual energy supplies to wireless networks. In practice, WPT is implementable by various technologies such as inductive coupling, magnetic resonate coupling, and electromagnetic (EM) radiation, for short-/mid-/long-range applications, respectively. In this paper, we consider the EM or radio signal enabled WPT in particular. Since radio signals can carry energy as well as information at the same time, a unified study on simultaneous wireless information and power transfer (SWIPT) is pursued. Specifically, this paper studies a multiple-input multiple-output (MIMO) wireless broadcast system consisting of three nodes, where one receiver harvests energy and another receiver decodes information separately from the signals sent by a common transmitter, and all the transmitter and receivers may be equipped with multiple antennas. Two scenarios are examined, in which the information receiver and energy receiver are separated and see different MIMO channels from the transmitter, or co-located and see the identical MIMO channel from the transmitter. For the case of separated receivers, we derive the optimal transmission strategy to achieve different tradeoffs for maximal information rate versus energy transfer, which are characterized by the boundary of a so-called rate-energy (R-E) region. For the case of co-located receivers, we show an outer bound for the achievable R-E region due to the potential limitation that practical energy harvesting receivers are not yet able to decode information directly. Under this constraint, we investigate two practical designs for the co-located receiver case, namely time switching and power splitting, and characterize their achievable R-E regions in comparison to the outer bound.
Strong Secrecy from Channel Resolvability
Venue: arXiv cs.IT 2013
Year: 2013

We analyze physical-layer security based on the premise that the coding mechanism for secrecy over noisy channels is tied to the notion of channel resolvability. Instead of considering capacity-based constructions, which associate to each message a sub-code that operates just below the capacity of the eavesdropper's channel, we consider channel-resolvability-based constructions, which associate to each message a sub-code that operates just above the resolvability of the eavesdropper's channel. Building upon the work of Csiszar and Hayashi, we provide further evidence that channel resolvability is a powerful and versatile coding mechanism for secrecy by developing results that hold for strong secrecy metrics and arbitrary channels. Specifically, we show that at least for symmetric wiretap channels, random capacity-based constructions fail to achieve the strong secrecy capacity while channel-resolvability-based constructions achieve it. We then leverage channel resolvability to establish the secrecy-capacity region of arbitrary broadcast channels with confidential messages and a cost constraint for strong secrecy metrics. Finally, we specialize our results to study the secrecy capacity of wireless channels with perfect channel state information, mixed channels and compound channels with receiver Channel State Information (CSI), as well as the secret-key capacity of source models for secret-key agreement. By tying secrecy to channel resolvability, we obtain achievable rates for strong secrecy metrics with simple proofs.
Bisimulations Meet PCTL Equivalences for Probabilistic Automata
Venue: arXiv cs.LO 2013
Year: 2013

Probabilistic automata (PAs) have been successfully applied in formal verification of concurrent and stochastic systems. Efficient model checking algorithms have been studied, where the most often used logics for expressing properties are based on probabilistic computation tree logic (PCTL) and its extension PCTL^*. Various behavioral equivalences are proposed, as a powerful tool for abstraction and compositional minimization for PAs. Unfortunately, the equivalences are well-known to be sound, but not complete with respect to the logical equivalences induced by PCTL or PCTL*. The desire of a both sound and complete behavioral equivalence has been pointed out by Segala in 1995, but remains open throughout the years. In this paper we introduce novel notions of strong bisimulation relations, which characterize PCTL and PCTL* exactly. We extend weak bisimulations that characterize PCTL and PCTL* without next operator, respectively. Further, we also extend the framework to simulation preorders. Thus, our paper bridges the gap between logical and behavioral equivalences and preorders in this setting.
Type classes for efficient exact real arithmetic in Coq
Venue: arXiv cs.LO 2013
Year: 2013

Floating point operations are fast, but require continuous effort on the part of the user in order to ensure that the results are correct. This burden can be shifted away from the user by providing a library of exact analysis in which the computer handles the error estimates. Previously, we [Krebbers/Spitters 2011] provided a fast implementation of the exact real numbers in the Coq proof assistant. Our implementation improved on an earlier implementation by O'Connor by using type classes to describe an abstract specification of the underlying dense set from which the real numbers are built. In particular, we used dyadic rationals built from Coq's machine integers to obtain a 100 times speed up of the basic operations already. This article is a substantially expanded version of [Krebbers/Spitters 2011] in which the implementation is extended in the various ways. First, we implement and verify the sine and cosine function. Secondly, we create an additional implementation of the dense set based on Coq's fast rational numbers. Thirdly, we extend the hierarchy to capture order on undecidable structures, while it was limited to decidable structures before. This hierarchy, based on type classes, allows us to share theory on the naturals, integers, rationals, dyadics, and reals in a convenient way. Finally, we obtain another dramatic speed-up by avoiding evaluation of termination proofs at runtime.
Source-Channel Coding Theorems for the Multiple-Access Relay Channel
Venue: arXiv cs.IT 2013
Year: 2013

We study reliable transmission of arbitrarily correlated sources over multiple-access relay channels (MARCs) and multiple-access broadcast relay channels (MABRCs). In MARCs only the destination is interested in reconstructing the sources, while in MABRCs both the relay and the destination want to reconstruct them. In addition to arbitrary correlation among the source signals at the users, both the relay and the destination have side information correlated with the source signals. Our objective is to determine whether a given pair of sources can be losslessly transmitted to the destination for a given number of channel symbols per source sample, defined as the source-channel rate. Sufficient conditions for reliable communication based on operational separation, as well as necessary conditions on the achievable source-channel rates are characterized. Since operational separation is generally not optimal for MARCs and MABRCs, sufficient conditions for reliable communication using joint source-channel coding schemes based on a combination of the correlation preserving mapping technique with Slepian-Wolf source coding are also derived. For correlated sources transmitted over fading Gaussian MARCs and MABRCs, we present conditions under which separation (i.e., separate and stand-alone source and channel codes) is optimal. This is the first time optimality of separation is proved for MARCs and MABRCs.
Some Properties of Rényi Entropy over Countably Infinite Alphabets
Venue: arXiv cs.IT 2013
Year: 2013

In this paper we study certain properties of Rényi entropy functionals $H_α(\mathcal{P})$ on the space of probability distributions over $\mathbb{Z}_+$. Primarily, continuity and convergence issues are addressed. Some properties shown parallel those known in the finite alphabet case, while others illustrate a quite different behaviour of Rényi entropy in the infinite case. In particular, it is shown that, for any distribution $\mathcal P$ and any $r\in[0,\infty]$, there exists a sequence of distributions $\mathcal{P}_n$ converging to $\mathcal{P}$ with respect to the total variation distance, such that $\lim_{n\to\infty}\lim_{α\to{1+}} H_α(\mathcal{P}_n) = \lim_{α\to{1+}}\lim_{n\to\infty} H_α(\mathcal{P}_n) + r$.
Achievable Outage Rate Regions for the MISO Interference Channel
Venue: arXiv cs.IT 2013
Year: 2013

We consider the slow-fading two-user multiple-input single-output (MISO) interference channel. We want to understand which rate points can be achieved, allowing a non-zero outage probability. We do so by defining four different outage rate regions. The definitions differ on whether the rates are declared in outage jointly or individually and whether the transmitters have instantaneous or statistical channel state information (CSI). The focus is on the instantaneous CSI case with individual outage, where we propose a stochastic mapping from the rate point and the channel realization to the beamforming vectors. A major contribution is that we prove that the stochastic component of this mapping is independent of the actual channel realization.
Algorithms for Unipolar and Generalized Split Graphs
Venue: arXiv cs.DM 2013
Year: 2013

A graph $G=(V,E)$ is a {\it unipolar graph} if there exits a partition $V=V_1 \cup V_2$ such that, $V_1$ is a clique and $V_2$ induces the disjoint union of cliques. The complement-closed class of {\it generalized split graphs} are those graphs $G$ such that either $G$ {\it or} the complement of $G$ is unipolar. Generalized split graphs are a large subclass of perfect graphs. In fact, it has been shown that almost all $C_5$-free (and hence, almost all perfect graphs) are generalized split graphs. In this paper we present a recognition algorithm for unipolar graphs that utilizes a minimal triangulation of the given graph, and produces a partition when one exists. Our algorithm has running time O($nm^\prime$), where $m^\prime$ is the number of edges in a minimal triangulation of the given graph. Generalized split graphs can recognized via this algorithm in O($nm' + n\OL{m}'$) = O($n^3$) time. We give algorithms on unipolar graphs for finding a maximum independent set and a minimum clique cover in O($n+m$) time and for finding a maximum clique and a minimum proper coloring in O($n^{2.5}/\log n$), when a unipolar partition is given. These algorithms yield algorithms for the four optimization problems on generalized split graphs that have the same worst-case time bound. We also prove that the perfect code problem is NP-Complete for unipolar graphs.