Unambiguous Parity‐Query Complexity

The approximate range emptiness problem requires a memory-efficient data structure D to approximately represent a set S of n distinct elements chosen from a large universe U= {0,1,⋯,N-1} and answer an emptiness query of the form “S∩[a;b]=0?” for an interval [a;b] of length L (a,b∈U), with a false positive rate e. The designed D for this problem can be kept in high-speed memory and quickly determine approximately whether a query interval is empty or not. Thus, it is crucial for facilitating online query processing in the information-centric Internet of Things applications, where the IoT data are continuously generated from a large number of resource-constrained sensors or readers and then are processed in networks. However, the existing works on the approximate range emptiness problem only consider the simple case when the set S is static, rendering them unsuitable for the continuously generated IoT data. In this paper, we study the approximate range emptiness problem over sliding windows in the IoT Data streams, denoted by e-ARESD-problem, where both insertion and deletion are allowed. We first prove that, given a sliding window size n and an interval length L, the lower bound of memory bits needed in any data structure for e-ARESD-problem is n log 2 (nL/e)+Θ(n). Then, a data structure is proposed and proved to be within a factor of 1.33 of the lower bound. The extensive simulation results demonstrate the advantage of the efficiency of our data structure over the baseline approach.

Answering range queries is a problem of fundamental importance in spatial information retrieval and computational geometry. The objective is to store a set of n points P in R d, each associated with a weight, so that it is possible to count, or more generally to compute some function of the weights of the points lying inside a given query range. Range searching is among the most heavily studied problems, and many search structures have been proposed and analyzed [1, 7]. There is a spectrum of space-time tradeoffs. The most relevant work to ours involves halfspace range counting queries, which Matouˇsek [6] has shown can be answered in n/m 1/d time from a data structure of space O(m). Nearly matching lower bounds were given by by Brönnimann, Chazelle and Pach [4] (or BCP). Given the relatively high complexity of range searching, it is natural to consider the problem in the context of approximation. We are given an approximation parameter ε> 0 and assume that ranges are bounded. Let η denote a range, and let diam(η) denote its diameter. All the points that lie in the range must be counted, and any of the points that lie within distance ε · diam(η) of the range’s boundary may be counted as well. Arya and Mount [3] showed that in any fixed dimension d with O(n log n) preprocessing time and O(n) space, ε-approximate range queries for any bounded convex range can be answered in time O(log n+1/ε d−1) [3]. Later, Chazelle, Liu, and Magen [5] considered approximate halfspace range and Euclidean ball searching in the high dimensional setting. Ignoring polylogarithmic factors, they showed that is possible to answer queries in O(d/ε 2) time with O(dn O(1/ε2) ) space.

Approximate sorting and its applications in I/O model

Computing shortest paths is one of the central problems in the theory of distributed computing. For the last few years, substantial progress has been made on the approximate single source shortest paths problem, culminating in an algorithm of Henzinger, Krinninger, and Nanongkai [STOC’16] which deterministically computes (1+o(1))-approximate shortest paths in Õ(D+√n) time, where D is the hop-diameter of the graph. Up to logarithmic factors, this time complexity is optimal, matching the lower bound of Elkin [STOC’04].

For a graph G=(V,E), finding a set of disjoint edges that do not share any vertices is called a matching problem, and finding the maximum matching is a fundamental problem in the theory of distributed graph algorithms. Although local algorithms for the approximate maximum matching problem have been widely studied, exact algorithms has not been much studied. In fact, no exact maximum matching algorithm that is faster than the trivial upper bound of O(n^2) rounds is known for the general instance. In this paper, we propose a randomized O(s_{max}^{3/2}+log n)-round algorithm in the CONGEST model, where s_{\max} is the size of maximum matching. This is the first exact maximum matching algorithm in o(n^2) rounds for general instances in the CONGEST model. The key technical ingredient of our result is a distributed algorithms of finding an augmenting path in O(s_{\max}) rounds, which is based on a novel technique of constructing a sparse certificate of augmenting paths, which is a subgraph of the input graph preserving at least one augmenting path. To establish a highly parallel construction of sparse certificates, we also propose a new characterization of sparse certificates, which might also be of independent interest.

Stochastic geometric optimization with joint probabilistic constraints

Stochastic geometric programming with joint probabilistic constraints

The polynomial approach introduced in Fuhrmann [1991] is extended to cover the crucial area of AAK theory, namely the characterization of zero location of the Schmidt vectors of the Hankel operators. This is done using the duality theory developed in that paper but with a twist. First we get the standard, lower bound, estimates on the number of unstable zeroes of the minimal degree Schmidt vectors of the Hankel operator. In the case of the Schmidt vector corresponding to the smallest singular the lower bound is in fact achieved. This leads to a solution of a Bezout equation. We use this Bezout equation to introduce another Hankel operator which have singular values that are the inverse of the singular values of the original Hankel operator.

A reproducing kernel Hilbert space (RKHS) approximation problem arising from learning theory is investigated. Some K-functionals and moduli of smoothness with respect to RKHSs are defined with Fourier–Bessel series and Fourier–Bessel transforms, respectively. Their equivalent relation is shown, with which the upper bound estimate for the best RKHS approximation is provided. The convergence rate is bounded with the defined modulus of smoothness, which shows that the RKHS approximation can attain the same approximation ability as that of the Fourier–Bessel series and Fourier–Bessel transform. In particular, it is shown that for a RKHS produced by the Bessel operator, the convergence rate sums up to the bound of a corresponding convolution operator approximation. The investigations show some new applications of Bessel functions. The results obtained can be used to bound the approximation error in learning theory.

We investigate the distance properties of linear locally recoverable codes (LRC codes) with all-symbol locality and availability. New upper and lower bounds on the minimum distance of such codes are derived. The upper bound is based on the shortening method and generalized Hamming weights that are fundamental parameters of any linear codes with many useful applications. This bound improves existing upper bounds. To reduce the gap in between upper and lower bounds, we do not restrict the alphabet size and propose explicit constructions of codes with locality and availability via rank-metric codes. The first construction relies on expander graphs and is better in low rate region. The second construction utilizes the LRC codes developed by Wang et al. as inner codes and is better in high rate region. We also suggest one possible generalization of LRC codes in which the recovering sets can intersect in a small number of coordinates. This feature allows us to increase the achievable code rate and still meet load balancing requirements. We derive upper and lower bounds on the parameters of such codes and present explicit constructions of codes with such a property.

A charge-based fixed-weight neural Hamming classifier with an on-chip normalization facility is described. The classifier utilizes a purely capacitive synapse matrix for quantization and a multiport sense amplifier for discrimination. The discriminator is compatible with variable-weight synapses as well. A detailed analysis of the classifier configuration is presented; design issues are addressed, and limitations are identified. It is shown that the ratio of the maximum Hamming weight to the minimum Hamming distance that can be handled by the classifier has an upper bound. As long as the exemplars comply with this upper bound, the network does not impose any limitation on the word length. A very large exemplar count, on the other hand, can impair connection density, but this problem can be averted by using multiple discriminators. A 2- mu m p-well CMOS test chip containing a Hamming classifier of ten 20-b-long exemplars is described.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

View-obstruction problems in n-dimensional geometry

The mixed (H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> /H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> ) control problem can be motivated as a nomninal LQG optimal control problem, subject to robust stability constraints, expressed in the form of an H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> norm bound. A related modified problem consisting on minimizing an upper bound of the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> cost subject to H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> constraints was introduced in [1]. Although there presently exist efficient methods to solve this modified problem, the original problem remains, to a large extent, still open. In this paper we propose a method for solving general discrete-time SISO (H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> /H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> ) problems. This method involves solving a sequence of problems, each one consisting of a finite-dimensional convex optimization and an unconstrained Nehari approximation problem

A new multivariate spectral estimation technique is proposed. It is based on a constrained spectrum approximation problem, where the distance between spectra is derived from the relative entropy rate between stationary Gaussian processes. This approach may be viewed as an extension of the high-resolution estimator called THREE introduced by Byrnes, Georgiou and Lindquist in 2000. The corresponding solution features a complexity upper bound which is equal to the one featured by THREE in the scalar case thereby improving on the one so far available in the multichannel framework. The solution is computed by means of a globally convergent, matricial Newton-type algorithm. Comparative simulation indicates that this new technique outperforms PEM and N4SID in the case of short data records.