An Algorithmic Toolbox for Network Calculus

Network Calculus offers powerful tools to analyze the performances in communication networks, in particular to obtain deterministic bounds. This theory is based on a strong mathematical ground, notably by the use of (min,+) algebra. However the algorithmic aspects of this theory have not been much addressed yet. This paper is an attempt to provide some efficient algorithms implementing Network Calculus operations for some classical functions. Some functions which are often used are the piecewise affine functions which ultimately have a constant growth. As a first step towards algorithmic design, we present a class containing these functions and closed under the Network Calculus operations: the piecewise affine functions which are ultimately pseudo-periodic. They can be finitely described which enables us to propose some algorithms for each of the Network Calculus operations. We finally analyze their computational complexity

Algebraic and algorithmic frameworks for optimized quantum measurements

Von Neumann projections are the main operations by which information can be extracted from the quantum to the classical realm. They are however static processes that do not adapt to the states they measure. Advances in the field of adaptive measurement have shown that this limitation can be overcome by "wrapping" the von Neumann projectors in a higher-dimensional circuit which exploits the interplay between measurement outcomes and measurement settings. Unfortunately, the design of adaptive measurement has often been ad hoc and setup-specific. We shall here develop a unified framework for designing optimized measurements. Our approach is two-fold: The first is algebraic and formulates the problem of measurement as a simple matrix diagonalization problem. The second is algorithmic and models the optimal interaction between measurement outcomes and measurement settings as a cascaded network of conditional probabilities. Finally, we demonstrate that several figures of merit, such as Bell factors, can be improved by optimized measurements. This leads us to the promising observation that measurement detectors which---taken individually---have a low quantum efficiency can be be arranged into circuits where, collectively, the limitations of inefficiency are compensated for

Isospeed: Improving (min,+) Convolution by Exploiting (min,+)/(max,+) Isomorphism

(min,+) convolution is the key operation in (min,+) algebra, a theory often used to compute performance bounds in real-time systems. As already observed in many works, its algorithm can be computationally expensive, due to the fact that: i) its complexity is superquadratic with respect to the size of the operands; ii) operands must be extended before starting its computation, and iii) said extension is tied to the least common multiple of the operand periods. In this paper, we leverage the isomorphism between (min,+) and (max,+) algebras to devise a new algorithm for (min,+) convolution, in which the need for operand extension is minimized. This algorithm is considerably faster than the ones known so far, and it allows us to reduce the computation times of (min,+) convolution by orders of magnitude

Computation of multi-degree B-splines

Multi-degree splines are smooth piecewise-polynomial functions where the pieces can have different degrees. We describe a simple algorithmic construction of a set of basis functions for the space of multi-degree splines, with similar properties to standard B-splines. These basis functions are called multi-degree B-splines (or MDB-splines). The construction relies on an extraction operator that represents all MDB-splines as linear combinations of local B-splines of different degrees. This enables the use of existing efficient algorithms for B-spline evaluations and refinements in the context of multi-degree splines. A Matlab implementation is provided to illustrate the computation and use of MDB-splines

Sum-of-squares of polynomials approach to nonlinear stability of fluid flows: an example of application

With the goal of providing the first example of application of a recently proposed method, thus demonstrating its ability to give results in principle, global stability of a version of the rotating Couette flow is examined. The flow depends on the Reynolds number and a parameter characterising the magnitude of the Coriolis force. By converting the original Navier-Stokes equations to a finite-dimensional uncertain dynamical system using a partial Galerkin expansion, high-degree polynomial Lyapunov functionals were found by sum-of-squares-of-polynomials optimization. It is demonstrated that the proposed method allows obtaining the exact global stability limit for this flow in a range of values of the parameter characterising the Coriolis force. Outside this range a lower bound for the global stability limit was obtained, which is still better than the energy stability limit. In the course of the study several results meaningful in the context of the method used were also obtained. Overall, the results obtained demonstrate the applicability of the recently proposed approach to global stability of the fluid flows. To the best of our knowledge, it is the first case in which global stability of a fluid flow has been proved by a generic method for the value of a Reynolds number greater than that which could be achieved with the energy stability approach

Efficient Enumeration of Non-Equivalent Squares in Partial Words with Few Holes

International audienceA partial word is a word with holes (also called don't cares: special symbols which match any symbol). A p-square is a partial word matching at least one standard square without holes (called a full square). Two p-squares are called equivalent if they match the same sets of full squares. Denote by psquares(T) the number of non-equivalent p-squares which are subwords of a partial word T. Let PSQUARES k (n) be the maximum value of psquares(T) over all partial words of length n with k holes. We show asympthotically tight bounds: c1 · min(nk 2 , n 2) ≤ PSQUARES k (n) ≤ c2 · min(nk 2 , n 2) for some constants c1, c2 > 0. We also present an algorithm that computes psquares(T) in O(nk 3) time for a partial word T of length n with k holes. In particular, our algorithm runs in linear time for k = O(1) and its time complexity near-matches the maximum number of non-equivalent p-squares

A very low latency pitch tracker for audio to midi conversion

International audienceAn algorithm for estimating the fundamental frequency of a single-pitch audio signal is described, for application to audio-to-MIDI conversion. In order to minimize latency, this method is based on the ESPRIT algorithm, together with a statistical model for partials frequencies. It is tested on real guitar recordings and compared to the YIN estimator. We show that, in this particular context, both methods exhibit a similar accuracy but the periodicity measure, used for note segmentation, is much more stable with the ESPRIT-based algorithm. This allows to significantly reduce ghost notes. This method is also able to get very close to the theoretical mini-mum latency, i.e. the fundamental period of the lowest observable pitch. Furthermore, it appears that fast implementations can reach a reasonable complexity and could be compatible with real-time, although this is not tested is this study