Optimal Regular Expressions for Permutations

The permutation language P_n consists of all words that are permutations of a fixed alphabet of size n. Using divide-and-conquer, we construct a regular expression R_n that specifies P_n. We then give explicit bounds for the length of R_n, which we find to be 4^{n}n^{-(lg n)/4+Theta(1)}, and use these bounds to show that R_n has minimum size over all regular expressions specifying P_n

Distributed PCP Theorems for Hardness of Approximation in P

We present a new distributed model of probabilistically checkable proofs (PCP). A satisfying assignment $x \in \{0,1\}^n$ to a CNF formula $\varphi$ is shared between two parties, where Alice knows $x_1, \dots, x_{n/2}$, Bob knows $x_{n/2+1},\dots,x_n$, and both parties know $\varphi$. The goal is to have Alice and Bob jointly write a PCP that $x$ satisfies $\varphi$, while exchanging little or no information. Unfortunately, this model as-is does not allow for nontrivial query complexity. Instead, we focus on a non-deterministic variant, where the players are helped by Merlin, a third party who knows all of $x$. Using our framework, we obtain, for the first time, PCP-like reductions from the Strong Exponential Time Hypothesis (SETH) to approximation problems in P. In particular, under SETH we show that there are no truly-subquadratic approximation algorithms for Bichromatic Maximum Inner Product over {0,1}-vectors, Bichromatic LCS Closest Pair over permutations, Approximate Regular Expression Matching, and Diameter in Product Metric. All our inapproximability factors are nearly-tight. In particular, for the first two problems we obtain nearly-polynomial factors of $2^{(\log n)^{1-o(1)}}$; only $(1+o(1))$-factor lower bounds (under SETH) were known before

On the symbol error probability of regular polytopes

An exact expression for the symbol error probability of the four-dimensional 24-cell in Gaussian noise is derived. Corresponding expressions for other regular convex polytopes are summarized. Numerically stable versions of these error probabilities are also obtained

On the distance between the expressions of a permutation

We prove that the combinatorial distance between any two reduced expressions of a given permutation of {1, ..., n} in terms of transpositions lies in O(n^4), a sharp bound. Using a connection with the intersection numbers of certain curves in van Kampen diagrams, we prove that this bound is sharp, and give a practical criterion for proving that the derivations provided by the reversing algorithm of [Dehornoy, JPAA 116 (1997) 115-197] are optimal. We also show the existence of length l expressions whose reversing requires C l^4 elementary steps

On the regularity of the covariance matrix of a discretized scalar field on the sphere

We present a comprehensive study of the regularity of the covariance matrix of a discretized field on the sphere. In a particular situation, the rank of the matrix depends on the number of pixels, the number of spherical harmonics, the symmetries of the pixelization scheme and the presence of a mask. Taking into account the above mentioned components, we provide analytical expressions that constrain the rank of the matrix. They are obtained by expanding the determinant of the covariance matrix as a sum of determinants of matrices made up of spherical harmonics. We investigate these constraints for five different pixelizations that have been used in the context of Cosmic Microwave Background (CMB) data analysis: Cube, Icosahedron, Igloo, GLESP and HEALPix, finding that, at least in the considered cases, the HEALPix pixelization tends to provide a covariance matrix with a rank closer to the maximum expected theoretical value than the other pixelizations. The effect of the propagation of numerical errors in the regularity of the covariance matrix is also studied for different computational precisions, as well as the effect of adding a certain level of noise in order to regularize the matrix. In addition, we investigate the application of the previous results to a particular example that requires the inversion of the covariance matrix: the estimation of the CMB temperature power spectrum through the Quadratic Maximum Likelihood algorithm. Finally, some general considerations in order to achieve a regular covariance matrix are also presented.Comment: 36 pages, 12 figures; minor changes in the text, matches published versio

Correlation function for the Grid-Poisson Euclidean matching on a line and on a circle

We compute the two-point correlation function for spin configurations which are obtained by solving the Euclidean matching problem, for one family of points on a grid, and the second family chosen uniformly at random, when the cost depends on a power $p$ of the Euclidean distance. We provide the analytic solution in the thermodynamic limit, in a number of cases ($p>1$ open b.c.\ and $p=2$ periodic b.c., both at criticality), and analyse numerically other parts of the phase diagram.Comment: 34 pages, 10 figure

NETEMBED: A Network Resource Mapping Service for Distributed Applications

Emerging configurable infrastructures such as large-scale overlays and grids, distributed testbeds, and sensor networks comprise diverse sets of available computing resources (e.g., CPU and OS capabilities and memory constraints) and network conditions (e.g., link delay, bandwidth, loss rate, and jitter) whose characteristics are both complex and time-varying. At the same time, distributed applications to be deployed on these infrastructures exhibit increasingly complex constraints and requirements on resources they wish to utilize. Examples include selecting nodes and links to schedule an overlay multicast file transfer across the Grid, or embedding a network experiment with specific resource constraints in a distributed testbed such as PlanetLab. Thus, a common problem facing the efficient deployment of distributed applications on these infrastructures is that of "mapping" application-level requirements onto the network in such a manner that the requirements of the application are realized, assuming that the underlying characteristics of the network are known. We refer to this problem as the network embedding problem. In this paper, we propose a new approach to tackle this combinatorially-hard problem. Thanks to a number of heuristics, our approach greatly improves performance and scalability over previously existing techniques. It does so by pruning large portions of the search space without overlooking any valid embedding. We present a construction that allows a compact representation of candidate embeddings, which is maintained by carefully controlling the order via which candidate mappings are inserted and invalid mappings are removed. We present an implementation of our proposed technique, which we call NETEMBED – a service that identify feasible mappings of a virtual network configuration (the query network) to an existing real infrastructure or testbed (the hosting network). We present results of extensive performance evaluation experiments of NETEMBED using several combinations of real and synthetic network topologies. Our results show that our NETEMBED service is quite effective in identifying one (or all) possible embeddings for quite sizable queries and hosting networks – much larger than what any of the existing techniques or services are able to handle.National Science Foundation (CNS Cybertrust 0524477, NSF CNS NeTS 0520166, NSF CNS ITR 0205294, EIA RI 0202067

Aberration in qualitative multilevel designs

Generalized Word Length Pattern (GWLP) is an important and widely-used tool for comparing fractional factorial designs. We consider qualitative factors, and we code their levels using the roots of the unity. We write the GWLP of a fraction ${\mathcal F}$ using the polynomial indicator function, whose coefficients encode many properties of the fraction. We show that the coefficient of a simple or interaction term can be written using the counts of its levels. This apparently simple remark leads to major consequence, including a convolution formula for the counts. We also show that the mean aberration of a term over the permutation of its levels provides a connection with the variance of the level counts. Moreover, using mean aberrations for symmetric $s^m$ designs with $s$ prime, we derive a new formula for computing the GWLP of ${\mathcal F}$. It is computationally easy, does not use complex numbers and also provides a clear way to interpret the GWLP. As case studies, we consider non-isomorphic orthogonal arrays that have the same GWLP. The different distributions of the mean aberrations suggest that they could be used as a further tool to discriminate between fractions.Comment: 16 pages, 1 figur