Log N Research Articles

Note on the Number of Hinges Defined by a Point Set in ℝ2

It is shown that the number of distinct types of three-point hinges, defined by a real plane set of n points is ≫n2 log−3n, where a hinge is identified by fixing two pairwise distances in a point triple. This is achieved via strengthening (modulo a logn factor) of the Guth- Katz estimate for the number of pairwise intersections of lines in ℝ3, arising in the context of the plane Erdős distinct distance problem, to a second moment incidence estimate. This relies, in particular, on the generalisation of the Guth-Katz incidence bound by Solomon and Sharir.

Geometric Algorithms for Finding a Point in the Intersection of Balls

We consider a problem of finding a point in the intersection of n balls in the Euclidean space Em. For the case m = 2 we suggest two algorithms of the complexity O(n2 log n) and O(n3) operations, respectively. For the general case we suggest an exact polynomial recursive algorithm which uses the orthogonal transformation of the space Em.

Faster compression of patterns to Rectangle Rule Lists

Access Control Lists (ACLs) are an essential security component in network routers. ACLs can be geometrically modeled as a two-dimensional black and white grid; our interest is in the most efficient way to represent such a grid. The more general problem is that of Rectangle Rule Lists (RRLs) minimization, which is finding the least number of rectangles needed to generate a given pattern. The scope of this paper focuses on a restricted version of RRLs minimization in which only rectangles that span the length or width of the grid are considered. Applegate et al.'s paper “Compressing Rectilinear Pictures and Minimizing Access Control Lists” gives an algorithm for finding an optimal solution for strip-rule RRLs minimization in O(n3) time, where n is the total number of rows and columns in the grid. Following the structure of Applegate et al.'s algorithm, we simplify the solution, remove redundancies in data structures, and exploit overlapping sub-problems in order to achieve an optimal solution for strip-rule RRLs minimization in O(n2 log n) time.

On the Law of the Iterated Logarithm Without Assumptions About the Existence of Moments

On the Law of the Iterated Logarithm Without Assumptions About the Existence of Moments

Cake-cutting with different entitlements: How many cuts are needed?

A cake has to be divided fairly among n agents. When all agents have equal entitlements, it is known that such a division can be implemented with n−1 cuts. When agents may have different entitlements, the paper shows that at least 2n−2 cuts may be necessary, and O(nlog⁡n) cuts are always sufficient.

The critical greedy server on the integers is recurrent

Each site of $\mathbb{Z}$ hosts a queue with arrival rate $\lambda$. A single server, starting at the origin, serves its current queue at rate $\mu$ until that queue is empty, and then moves to the longest neighbouring queue. In the critical case $\lambda = \mu$, we show that the server returns to every site infinitely often. We also give a sharp iterated logarithm result for the server's position. Important ingredients in the proofs are that the times between successive queues being emptied exhibit doubly exponential growth, and that the probability that the server changes its direction is asymptotically equal to $1/4$.

Using Petal-Decompositions to Build a Low Stretch Spanning Tree

We prove that any weighted graph $G=(V,E,w)$ with $n$ points and $m$ edges has a spanning tree $T$ such that $\sum_{\{u,v\}\in E}\frac{d_T(u,v)}{w(u,v)}=O(m\log n\log\log n)$. Moreover, such a tree...

Efficient Connected Dominating Set Construction with Maximum Lifetime in the Cognitive Radio Networks

Connected dominating set (CDS) is the most representative technology for constructing a virtual backbone in wireless networks and plays an important role in wireless applications including broadcasting, routing and so on. In a cognitive radio networks, the lifetime and efficiency are two important indices for measuring CDS algorithms due to the random activities of primary users (PUs). However, to the best of our knowledge, the existing algorithms for CDS construction in CRNs ignore the execution effectiveness instead of lifetime. In this paper, we propose a four-phase distributed algorithm to maximize the lifetime of CDS while guaranteeing the effectiveness of the algorithm. The proposed algorithm terminates in O(N3 log N) timeslots, which is more effective than that of O(N4).

On a biased edge isoperimetric inequality for the discrete cube

The ‘full’ edge isoperimetric inequality for the discrete cube {0,1}n (due to Harper, Lindsey, Berstein and Hart) specifies the minimum size of the edge boundary ∂A of a set A⊂{0,1}n, as function of |A|. A weaker (but more widely-used) lower bound is |∂A|≥|A|log⁡(2n/|A|), where equality holds whenever A is a subcube. In 2011, the first author obtained a sharp ‘stability’ version of the latter result, proving that if |∂A|≤|A|(log⁡(2n/|A|)+ϵ), then there exists a subcube C such that |AΔC|/|A|=O(ϵ/log⁡(1/ϵ)).The ‘weak’ version of the edge isoperimetric inequality has the following well-known generalization for the ‘p-biased’ measure μp on the discrete cube: if p≤1/2, or if 0<p<1 and A is monotone increasing, then pμp(∂A)≥μp(A)logp⁡(μp(A)).In this paper, we prove a sharp stability version of the latter result, which generalizes the aforementioned result of the first author. Namely, we prove that if pμp(∂A)≤μp(A)(logp⁡(μp(A))+ϵ), then there exists a subcube C such that μp(AΔC)/μp(A)=O(ϵ′/log⁡(1/ϵ′)), where ϵ′:=ϵln⁡(1/p). This result is a central component in recent work of the authors proving sharp stability versions of a number of Erdős–Ko–Rado type theorems in extremal combinatorics, including the seminal ‘complete intersection theorem’ of Ahlswede and Khachatrian.In addition, we prove a biased-measure analogue of the ‘full’ edge isoperimetric inequality, for monotone increasing sets, and we observe that such an analogue does not hold for arbitrary sets, hence answering a question of Kalai. We use this result to give a new proof of the ‘full’ edge isoperimetric inequality, one relying on the Kruskal–Katona theorem.

Re-examination of Bregman functions and new properties of their divergences

ABSTRACTThe Bregman divergence (Bregman distance, Bregman measure of distance) is a certain useful substitute for a distance, obtained from a well-chosen function (the ‘Bregman function’). Bregman functions and divergences have been extensively investigated during the last decades and have found applications in optimization, operations research, information theory, nonlinear analysis, machine learning and more. This paper re-examines various aspects related to the theory of Bregman functions and divergences. In particular, it presents many sufficient conditions which allow the construction of Bregman functions in a general setting and introduces new Bregman functions (such as a negative iterated log entropy). Moreover, it sheds new light on several known Bregman functions such as quadratic entropies, the negative Havrda-Charvát-Tsallis entropy, and the negative Boltzmann-Gibbs-Shannon entropy, and it shows that the negative Burg entropy, which is not a Bregman function according to the classical theory but nevertheless is known to have ‘Bregmanian properties’, can, by our re-examination of the theory, be considered as a Bregman function. Our analysis yields several by-products of independent interest such as the introduction of the concept of relative uniform convexity (a certain generalization of uniform convexity), new properties of uniformly and strongly convex functions, and results in Banach space theory.

Polynomial expressions for non-binomial structures

Following recent paper of the author about polynomial expressions with respect to binomial ideals, the current paper is devoted to the non-binomial case. Beside the established facts in the underlying theory, the correctness and termination of the algorithms are addressed. From the collection of examples, the main large-scale non-binomial one is of particular importance. Application of the new algorithms for the cases of moderately large binomial like cyclic 16-roots and cyclic 18-roots is discussed. A combinatorial analysis for complexity of the proposed algorithms is shown to be at least O(log⁡(log⁡n)n2(log⁡n)2).

Moderate deviations for Euler-Maruyama approximation of Hull-White stochastic volatility model

We consider the Euler-Maruyama discretization of stochastic volatility model dSt = σtStdWt, dσt = ωσtdZt, t ∈ [0, T], which has been widely used in financial practice, where Wt,Zt, t ∈ [0, T], are two uncorrelated standard Brownian motions. Using asymptotic analysis techniques, the moderate deviation principles for log Sn (or log |Sn| in case Sn is negative) are obtained as n → ∞ under different discretization schemes for the asset price process St and the volatility process σt. Numerical simulations are presented to compare the convergence speeds in different schemes.

“Direct” Gas-phase Metallicity in Local Analogs of High-redshift Galaxies: Empirical Metallicity Calibrations for High-redshift Star-forming Galaxies

Abstract We study the direct gas-phase oxygen abundance using the well-detected auroral line [O iii]λ4363 in the stacked spectra of a sample of local analogs of high-redshift galaxies. These local analogs share the same location as z ∼ 2 star-forming galaxies on the [O iii]λ5007/Hβ versus [N ii]λ6584/Hα Baldwin–Phillips–Terlevich diagram. This type of analog has the same ionized interstellar medium (ISM) properties as high-redshift galaxies. We establish empirical metallicity calibrations between the direct gas-phase oxygen abundances ( ) and the N2 (log([N ii]λ6584/Hα))/O3N2 (log(([O iii]λ5007/Hβ)/([N ii]λ6584/Hα))) indices in our local analogs. We find significant systematic offsets between the metallicity calibrations for our local analogs of high-redshift galaxies and those derived from the local H ii regions and a sample of local reference galaxies selected from the Sloan Digital Sky Survey (SDSS). The N2 and O3N2 metallicities will be underestimated by 0.05–0.1 dex relative to our calibration, if one simply applies the local metallicity calibration in previous studies to high-redshift galaxies. Local metallicity calibrations also cause discrepancies of metallicity measurements in high-redshift galaxies using the N2 and O3N2 indicators. In contrast, our new calibrations produce consistent metallicities between these two indicators. We also derive metallicity calibrations for R23 (log(([O iii]λλ4959,5007+[O ii]λλ3726,3729)/Hβ)), O32(log([O iii]λλ4959,5007/[O ii]λλ3726,3729)), [O iii]λ5007/Hβ), and log([Ne iii]λ3869/[O ii]λ3727) indices in our local analogs, which show significant offset compared to those in the SDSS reference galaxies. By comparing with MAPPINGS photoionization models, the different empirical metallicity calibration relations in the local analogs and the SDSS reference galaxies can be shown to be primarily due to the change of ionized ISM conditions. Assuming that temperature structure variations are minimal and ISM conditions do not change dramatically from z ∼ 2 to z ∼ 5, these empirical calibrations can be used to measure relative metallicities in galaxies with redshifts up to z ∼ 5.0 in ground-based observations.

Structural Results on Matching Estimation with Applications to Streaming

We study the problem of estimating the size of a matching when the graph is revealed in a streaming fashion. Our results are multifold:

Threesomes, Degenerates, and Love Triangles

The 3SUM problem is to decide, given a set of n real numbers, whether any three sum to zero. It is widely conjectured that a trivial O ( n 2 )-time algorithm is optimal on the Real RAM, and optimal even in the nonuniform linear decision tree model. Over the years the consequences of this conjecture have been revealed. This 3SUM conjecture implies Ω ( n 2 ) lower bounds on numerous problems in computational geometry, and a variant of the conjecture for integer inputs implies strong lower bounds on triangle enumeration, dynamic graph algorithms, and string matching data structures. In this article, we refute the conjecture that 3SUM requires Ω ( n 2 ) in the Real RAM and refute more forcefully the conjecture that its complexity is Ω ( n 2 ) in the linear decision tree model. In particular, we prove that the decision tree complexity of 3SUM is O ( n 3/2 √ log n ) and give two subquadratic 3SUM algorithms, a deterministic one running in O ( n 2 / (log n / log log n ) 2/3 ) time and a randomized one running in O ( n 2 (log log n ) 2 / log n ) time with high probability. Our results lead directly to improved bounds on the decision tree complexity of k -variate linear degeneracy testing for all odd k ≥ 3. Finally, we give a subcubic algorithm for a generalization of the (min ,+)-product over real-valued matrices and apply it to the problem of finding zero-weight triangles in edge-weighted graphs. We give a depth- O ( n 5/2 √ log n ) decision tree for this problem, as well as a deterministic algorithm running in time O ( n 3 (log log n ) 2 /log n ).

Fast generation of isotropic Gaussian random fields on the sphere

Abstract The efficient simulation of isotropic Gaussian random fields on the unit sphere is a task encountered frequently in numerical applications. A fast algorithm based on Markov properties and fast Fourier transforms in 1d is presented that generates samples on an n × n {n\times n} grid in O ⁡ ( n 2 ⁢ log ⁡ n ) {\operatorname{O}(n^{2}\log n)} . Furthermore, an efficient method to set up the necessary conditional covariance matrices is derived and simulations demonstrate the performance of the algorithm. An open source implementation of the code has been made available at https://github.com/pec27/smerfs.

Discordant Voting Processes on Finite Graphs

We consider an asynchronous voting process on graphs which we call discordant voting, and which can be described as follows. Initially each vertex holds one of two opinions, red or blue say. Neighbouring vertices with different opinions interact pairwise. After an interaction both vertices have the same colour. The quantity of interest is T, the time to reach consensus, i.e. the number of interactions needed for all vertices have the same colour. An edge whose endpoint colours differ (i.e. one vertex is coloured red and the other one blue) is said to be discordant. A vertex is discordant if its is incident with a discordant edge. In discordant voting, all interactions are based on discordant edges. Because the voting process is asynchronous there are several ways to update the colours of the interacting vertices. Push: Pick a random discordant vertex and push its colour to a random discordant neighbour. Pull: Pick a random discordant vertex and pull the colour of a random discordant neighbour. Oblivious: Pick a random endpoint of a random discordant edge and push the colour to the other end point. We show that ET, the expected time to reach consensus, depends strongly on the underlying graph and the update rule. For connected graphs on n vertices, and an initial half red, half blue colouring the following hold. For oblivious voting, ET = n2/4 independent of the underlying graph. For the complete graph Kn, the push protocol has ET = =(n log n), whereas the pull protocol has ET = =(2n). For the cycle Cn all three protocols have ET = =(n2). For the star graph however, the pull protocol has ET = O(n2), whereas the push protocol is slower with ET = =(n2 log n). The wide variation in ET for the pull protocol is to be contrasted with the well known model of synchronous pull voting, for which ET = O(n) on many classes of expanders.

A minimum knapsack-based resource allocation for underlaying device-to-device communication

As the numbers of smart devices increases, proximity-based services have enabled device-to-device (D2D) communication to be regarded as one of the major communication paradigms. In underlaying D2D, the devices communicating with each other directly use shared cellular resources. In this paper we propose a minimum knapsack-based interference aware resource allocation algorithm (MIKIRA) for D2D communication underlaying cellular networks. We compare the system sum rates, interference and signal-to-interference-and-noise-ratios (SINR) of MIKIRA with a graph-based resource allocation (GRA) algorithm and random allocation. In our three different sets of experiments with different percentages of D2D pairs in the total number of cellular users in the network, we observe that MIKIRA performs better than the other algorithms in terms of interference and SINR, and obtains a similar system sum rate. MIKIRA (O(n2 log(n))) is also computationally more efficient when compared with the GRA (O(n3)), which makes it suitable for use in the LTE scheduling period of 1 ms.

Coherent field propagation between tilted planes.

Propagating electromagnetic light fields between nonparallel planes is of special importance, e.g., within the design of novel computer-generated holograms or the simulation of optical systems. In contrast to the extensively discussed evaluation between parallel planes, the diffraction-based propagation of light onto a tilted plane is more burdensome, since discrete fast Fourier transforms cannot be applied directly. In this work, we propose a quasi-fast algorithm (O(N3 log N)) that deals with this problem. Based on a proper decomposition into three rotations, the vectorial field distribution is calculated on a tilted plane using the spectrum of plane waves. The algorithm works on equidistant grids, so neither nonuniform Fourier transforms nor an explicit complex interpolation is necessary. The proposed algorithm is discussed in detail and applied to several examples of practical interest.

Exact Rates in the Davis–Gut Law of Iterated Logarithm for the First-Moment Convergence of Independent Identically Distributed Random Variables

Let {X, X n , n ≥ 1} be a sequence of independent identically distributed random variables and let $$ {S}_n={\sum}_{i=1}^n{X}_i $$ , M n = max1 ≤ k ≤ n|S k |. For r > 0, let a n (e) be a function of e such that a n (e) log log n→τ as n→∞ and $$ \varepsilon \searrow \sqrt{r} $$ . If $$ \mathbb{E}{X}^2I\left\{\left|X\right|\ge t\right\}=o\left({\left(\log\;\log\;t\right)}^{-1}\right) $$ as t→∞, then, by using the strong approximation, we show that $$ \underset{\varepsilon \searrow \sqrt{r}}{\lim}\frac{1}{-\log \left({\varepsilon}^2-r\right)}\sum_{n=1}^{\infty}\frac{{\left(\log \kern0.5em n\right)}^{r-1}}{n^{3/2}}\mathbb{E}{\left\{{M}_n-\left(\varepsilon +{a}_n\left(\varepsilon \right)\right)\sigma \sqrt{2n\;\log\;\log\;n}\right\}}_{+}=\frac{2\sigma {e}^{-2\tau \sqrt{r}}}{\sqrt{2\pi }r} $$ holds if and only if $$ \mathbb{E}X=0 $$ , $$ \mathbb{E}{X}^2={\sigma}^2 $$ , and $$ \mathbb{E}{X}^2{\left(\log \left|X\right|\right)}^{r-1}{\left(\log\;\log\;\left|X\right|\right)}^{-\frac{1}{2}}<\infty $$ .