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The discrepancy of a binary string is the maximum (absolute) difference between the number of ones and the number of zeroes over all possible substrings of the given binary string. In this note we determine the minimal discrepancy that a binary de Bruijn sequence of order $n$ can achieve, which is $n$. This was an open problem until now. We give an algorithm that constructs a binary de Bruijn sequence with minimal discrepancy. A slight modification of this algorithm deals with arbitrary alphabets and yields de Bruijn sequences of order $n$ with discrepancy at most $1$ above the trivial lower bound $n$.

In 1972 Mykkeltveit proved that the maximum number of vertex-disjoint cycles in the de Bruijn graphs of order $n$ is attained by the pure cycling register rule, as conjectured by Golomb. We generalize this result to the tensor product of the de Bruijn graph of order $n$ and a simple cycle of size $k$, when $n$ divides $k$ or vice versa. We also develop counting formulae for a large family of cycling register rules, including the linear register rules proposed by Golomb.

We study small white noise perturbations of planar dynamical systems with heteroclinic networks in the limit of vanishing noise. We show that the probabilities of transitions between various cells that the network tessellates the plane into decay as powers of the noise magnitude. We show that the most likely scenario for the realization of these rare transition events involves spending atypically long times in the neighborhoods of certain saddle points of the network. We describe the hierarchy of time scales and clusters of accessibility associated with these rare transition events. We discuss applications of our results to homogenization problems and to the invariant distribution asymptotics. At the core of our results are local limit theorems for exit distributions obtained via methods of Malliavin calculus., Comment: 141 pages, 12 figures

For a one-dimensional smooth vector field in a neighborhood of an unstable equilibrium, we consider the associated dynamics perturbed by small noise. We give a revealing elementary proof of a result proved earlier using heavy machinery from Malliavin calculus. In particular, we obtain precise vanishing noise asymptotics for the tail of the exit time and for the exit distribution conditioned on atypically long exits., Comment: 20 pages; the main difference with the previous version is a new section on noisy heteroclinic networks which are the main motivation for the present study

For a one-dimensional smooth vector field in a neighborhood of an unstable equilibrium, we consider the associated dynamics perturbed by small noise. Using Malliavin calculus tools, we obtain precise vanishing noise asymptotics for the tail of the exit time and for the exit distribution conditioned on atypically long exits., Comment: 21 pages

We consider white noise perturbations of a nonlinear dynamical system in the neighborhood of an unstable critical point with linearization given by a Jordan block of full dimension. For the associated exit problem, we study the joint limiting behavior of the exit location and exit time, in the vanishing noise limit. The exit typically happens near one of two special deterministic points associated with the eigendirection, and we obtain several more terms in the expansion for the exit point. The leading correction term is deterministic and logarithmic in the noise magnitude, while the random remainder satisfies a scaling limit., Comment: 18 pages

We introduce the Connection Scan Algorithm (CSA) to efficiently answer queries to timetable information systems. The input consists, in the simplest setting, of a source position and a desired target position. The output consist is a sequence of vehicles such as trains or buses that a traveler should take to get from the source to the target. We study several problem variations such as the earliest arrival and profile problems. We present algorithm variants that only optimize the arrival time or additionally optimize the number of transfers in the Pareto sense. An advantage of CSA is that is can easily adjust to changes in the timetable, allowing the easy incorporation of known vehicle delays. We additionally introduce the Minimum Expected Arrival Time (MEAT) problem to handle possible, uncertain, future vehicle delays. We present a solution to the MEAT problem that is based upon CSA. Finally, we extend CSA using the multilevel overlay paradigm to answer complex queries on nation-wide integrated timetables with trains and buses.

This paper studies the intermediate time behaviour of a small random perturbation of a periodic cellular flow. Our main result shows that on time scales shorter than the diffusive time scale, the limiting behaviour of trajectories that start close enough to cell boundaries is a fractional kinetic process: A Brownian motion time changed by the local time of an independent Brownian motion. Our proof uses the Freidlin-Wentzell framework, and the key step is to establish an analogous averaging principle on shorter time scales. As a consequence of our main theorem, we obtain a homogenization result for the associated advection-diffusion equation. We show that on intermediate time scales the effective equation is a fractional time PDE that arises in modelling anomalous diffusion., Comment: 47 pages

A multifractal analysis (MFA) is performed on three-dimensional grayscale images associated with natural porous structures (soil samples). First, computed tomography (CT) scans are carried out on the samples to generate 3D grayscale images. Then, a preliminary analysis is conducted to evaluate key quantities associated with the porosity, such as void fraction, pore volume, connectivity, and surface area. Finally, the samples are successfully identified and separated into two different structure families by using the MFA. A new software (Munari) to carry out the MFA of 3D grayscale images is also presented., Comment: 11 pages, 2 tables, 3 figures

We show that the slicing problem holds true for subspaces of $L_p,p>2$ in the setting of arbitrary measures in place of volume. This generalizes a result of Milman for the original slicing problem.

There has been tremendous progress in algorithmic methods for computing driving directions on road networks. Most of that work focuses on time-independent route planning, where it is assumed that the cost on each arc is constant per query. In practice, the current traffic situation significantly influences the travel time on large parts of the road network, and it changes over the day. One can distinguish between traffic congestion that can be predicted using historical traffic data, and congestion due to unpredictable events, e.g., accidents. In this work, we study the \emph{dynamic and time-dependent} route planning problem, which takes both prediction (based on historical data) and live traffic into account. To this end, we propose a practical algorithm that, while robust to user preferences, is able to integrate global changes of the time-dependent metric~(e.g., due to traffic updates or user restrictions) faster than previous approaches, while allowing subsequent queries that enable interactive applications.

Let $A$ be a matrix whose columns $X_1,\dots, X_N$ are independent random vectors in $\mathbb{R}^n$. Assume that the tails of the 1-dimensional marginals decay as $\mathbb{P}(|\langle X_i, a\rangle|\geq t)\leq t^{-p}$ uniformly in $a\in S^{n-1}$ and $i\leq N$. Then for $p>4$ we prove that with high probability $A/{\sqrt{n}}$ has the Restricted Isometry Property (RIP) provided that Euclidean norms $|X_i|$ are concentrated around $\sqrt{n}$. We also show that the covariance matrix is well approximated by the empirical covariance matrix and establish corresponding quantitative estimates on the rate of convergence in terms of the ratio $n/N$. Moreover, we obtain sharp bounds for both problems when the decay is of the type $ \exp({-t^{\alpha}})$ with $\alpha \in (0,2]$, extending the known case $\alpha\in[1, 2]$., Comment: To appear in J. Eur. Math. Soc

We consider a perturbed ordinary differential equation where the perturbation is only significant when a one-dimensional null recurrent diffusion is close to zero. We investigate the first order correction to the unperturbed system and prove a central limit theorem type result, i.e., that the normalized deviation process converges in law in the space of continuous functions to a limit process which we identify. We show that this limit process has a component which only moves when the limit of the null-recurrent fast motion equals zero. The set of these times forms a zero-measure Cantor set and therefore the limiting process cannot be described by a standard SDE. We characterize this process by its infinitesimal generator (with appropriate boundary conditions) and we also characterize the process as the weak solution of an SDE that depends on the local time of the fast motion process. We also investigate the long time behavior of such a system when the unperturbed motion is trivial. In this case, we show that the long-time limit is constant on a set of full Lebesgue measure with probability 1, but it has nontrivial drift and diffusion components that move only when the fast motion equals zero., Comment: 37 pages

We study the journey planning problem in public transit networks. Developing efficient preprocessing-based speedup techniques for this problem has been challenging: current approaches either require massive preprocessing effort or provide limited speedups. Leveraging recent advances in Hub Labeling, the fastest algorithm for road networks, we revisit the well-known time-expanded model for public transit. Exploiting domain-specific properties, we provide simple and efficient algorithms for the earliest arrival, profile, and multicriteria problems, with queries that are orders of magnitude faster than the state of the art., Comment: An extended abstract of this paper has been accepted at the 14th International Symposium on Experimental Algorithms (SEA'15)

We survey recent advances in algorithms for route planning in transportation networks. For road networks, we show that one can compute driving directions in milliseconds or less even at continental scale. A variety of techniques provide different trade-offs between preprocessing effort, space requirements, and query time. Some algorithms can answer queries in a fraction of a microsecond, while others can deal efficiently with real-time traffic. Journey planning on public transportation systems, although conceptually similar, is a significantly harder problem due to its inherent time-dependent and multicriteria nature. Although exact algorithms are fast enough for interactive queries on metropolitan transit systems, dealing with continent-sized instances requires simplifications or heavy preprocessing. The multimodal route planning problem, which seeks journeys combining schedule-based transportation (buses, trains) with unrestricted modes (walking, driving), is even harder, relying on approximate solutions even for metropolitan inputs., Comment: This is an updated version of the technical report MSR-TR-2014-4, previously published by Microsoft Research. This work was mostly done while the authors Daniel Delling, Andrew Goldberg, and Renato F. Werneck were at Microsoft Research Silicon Valley

We present an effective heuristic for the Steiner Problem in Graphs. Its main elements are a multistart algorithm coupled with aggressive combination of elite solutions, both leveraging recently-proposed fast local searches. We also propose a fast implementation of a well-known dual ascent algorithm that not only makes our heuristics more robust (by quickly dealing with easier cases), but can also be used as a building block of an exact (branch-and-bound) algorithm that is quite effective for some inputs. On all graph classes we consider, our heuristic is competitive with (and sometimes more effective than) any previous approach with similar running times. It is also scalable: with long runs, we could improve or match the best published results for most open instances in the literature., Comment: 33 pages. Presented at the 11th DIMACS Implementation Challenge on Steiner Tree Problems

A premise at a heart of network analysis is that entities in a network derive utilities from their connections. The {\em influence} of a seed set $S$ of nodes is defined as the sum over nodes $u$ of the {\em utility} of $S$ to $u$. {\em Distance-based} utility, which is a decreasing function of the distance from $S$ to $u$, was explored in several successful research threads from social network analysis and economics: Network formation games [Bloch andJackson 2007], Reachability-based influence [Richardson and Domingos 2002, Kempe et al. 2003], "threshold" influence [Gomez-Rodriguez et al. 2011], and {\em closeness centrality} [Bavelas 1948]. We formulate a model that unifies and extends this previous work and address the two fundamental computational problems in this domain: {\em Influence oracles} and {\em influence maximization} (IM). An oracle performs some preprocessing, after which influence queries for arbitrary seed sets can be efficiently computed. With IM, we seek a set of nodes of a given size with maximum influence. Since the IM problem is computationally hard, we instead seek a {\em greedy sequence} of nodes, with each prefix having influence that is at least $1-1/e$ of that of the optimal seed set of the same size. We present the first highly scalable algorithms for both problems, providing statistical guarantees on approximation quality and near-linear worst-case bounds on the computation. We perform an experimental evaluation which demonstrates the effectiveness of our designs on networks with hundreds of millions of edges., Comment: 20 pages, 5 figures

An ordinary differential equation perturbed by a null-recurrent diffusion will be considered in the case where the averaging type perturbation is strong only when a fast motion is close to the origin. The normal deviations of these solutions from the averaged motion are studied, and a central limit type theorem is proved. The limit process satisfies a linear equation driven by a Brownian motion time changed by the local time of the fast motion., Comment: 15 pages, 1 figure Corollary 2.4 fixed

Closeness centrality, first considered by Bavelas (1948), is an importance measure of a node in a network which is based on the distances from the node to all other nodes. The classic definition, proposed by Bavelas (1950), Beauchamp (1965), and Sabidussi (1966), is (the inverse of) the average distance to all other nodes. We propose the first highly scalable (near linear-time processing and linear space overhead) algorithm for estimating, within a small relative error, the classic closeness centralities of all nodes in the graph. Our algorithm applies to undirected graphs, as well as for centrality computed with respect to round-trip distances in directed graphs. For directed graphs, we also propose an efficient algorithm that approximates generalizations of classic closeness centrality to outbound and inbound centralities. Although it does not provide worst-case theoretical approximation guarantees, it is designed to perform well on real networks. We perform extensive experiments on large networks, demonstrating high scalability and accuracy., Comment: 13 pages, 2 figures, appeared at the 2nd ACM Conference on Online Social Networks (COSN'14)

Propagation of contagion through networks is a fundamental process. It is used to model the spread of information, influence, or a viral infection. Diffusion patterns can be specified by a probabilistic model, such as Independent Cascade (IC), or captured by a set of representative traces. Basic computational problems in the study of diffusion are influence queries (determining the potency of a specified seed set of nodes) and Influence Maximization (identifying the most influential seed set of a given size). Answering each influence query involves many edge traversals, and does not scale when there are many queries on very large graphs. The gold standard for Influence Maximization is the greedy algorithm, which iteratively adds to the seed set a node maximizing the marginal gain in influence. Greedy has a guaranteed approximation ratio of at least (1-1/e) and actually produces a sequence of nodes, with each prefix having approximation guarantee with respect to the same-size optimum. Since Greedy does not scale well beyond a few million edges, for larger inputs one must currently use either heuristics or alternative algorithms designed for a pre-specified small seed set size. We develop a novel sketch-based design for influence computation. Our greedy Sketch-based Influence Maximization (SKIM) algorithm scales to graphs with billions of edges, with one to two orders of magnitude speedup over the best greedy methods. It still has a guaranteed approximation ratio, and in practice its quality nearly matches that of exact greedy. We also present influence oracles, which use linear-time preprocessing to generate a small sketch for each node, allowing the influence of any seed set to be quickly answered from the sketches of its nodes., Comment: 10 pages, 5 figures. Appeared at the 23rd Conference on Information and Knowledge Management (CIKM 2014) in Shanghai, China

We consider a two-parameter averaging-homogenization type elliptic problem together with the stochastic representation of the solution. A limit theorem is derived for the corresponding diffusion process and a precise description of the two-parameter limit behavior for the solution of the PDE is obtained.

We consider $n\times n$ random matrices $M_{n}=\sum_{\alpha =1}^{m}{\tau _{\alpha }}\mathbf{y}_{\alpha }\otimes \mathbf{y}_{\alpha }$, where $\tau _{\alpha }\in \mathbb{R}$, $\{\mathbf{y}_{\alpha }\}_{\alpha =1}^{m}$ are i.i.d. isotropic random vectors of $\mathbb{R}^n$, whose components are not necessarily independent. It was shown in arXiv:0710.1346 that if $m,n\rightarrow \infty$, $m/n\rightarrow c\in \lbrack 0,\infty )$, the Normalized Counting Measures of $\{\tau _{\alpha }\}_{\alpha =1}^{m}$ converge weakly and $\{\mathbf{y}_\alpha\}_{\alpha=1}^m$ are \textit{good} (see corresponding definition), then the Normalized Counting Measures of eigenvalues of $M_{n}$ converge weakly in probability to a non-random limit found in \cite{Ma-Pa:67}. In this paper we indicate a subclass of good vectors, which we call \textit{very good} and for which the linear eigenvalue statistics of the corresponding matrices converge in distribution to the Gaussian law, i.e., the Central Limit Theorem is valid. An important example of good vectors, studied in arXiv:0710.1346 are the vectors with log-concave distribution. We discuss the conditions for them, guaranteeing the validity of the Central Limit Theorem for linear eigenvalue statistics of corresponding matrices.

We prove that for $s<0$, $s$-concave measures on ${\mathbb R}^n$ satisfy a thin shell concentration similar to the log-concave one. It leads to a Berry-Esseen type estimate for their one dimensional marginal distributions. We also establish sharp reverse H\"older inequalities for $s$-concave measures.

The aim of this paper is to investigate the distribution of a continuous homopolymer in the presence of an attractive finitely supported potential. The most intricate behavior can be observed if we simultaneously vary two parameters: the temperature, which approaches the critical value, and the length of the polymer, which tends to infinity. As the main result, we identify the distributions that appear in the limit (after a diffusive scaling of the original polymer measures) and depend on the relation between the two parameters.

Let $p\geq 1$, $\eps >0$, $r\geq (1+\eps) p$, and $X$ be a $(-1/r)$-concave random vector in $\R^n$ with Euclidean norm $|X|$. We prove that $(\E |X|^{p})^{1/{p}}\leq c (C(\eps) \E|X|+\sigma_{p}(X))$, where $\sigma_{p}(X)=\sup_{|z|\leq 1}(\E||^{p})^{1/p}$, $C(\eps)$ depends only on $\eps$ and $c$ is a universal constant. Moreover, if in addition $X$ is centered then $(\E |X|^{-p})^{-1/{p}}\geq c(\eps) (\E|X| - C \sigma_{p}(X))$.

We give a short proof of a result of G. Paouris on the tail behaviour of the Euclidean norm $|X|$ of an isotropic log-concave random vector $X\in\R^n$, stating that for every $t\geq 1$, $P(|X|\geq ct\sqrt n)\leq \exp(-t\sqrt n)$. More precisely we show that for any log-concave random vector $X$ and any $p\geq 1$, $(E|X|^p)^{1/p}\sim E |X|+\sup_{z\in S^{n-1}}(E |< z,X>|^p)^{1/p}$.