A Linear-Size Logarithmic Stretch Path-Reporting Distance Oracle for General Graphs

In 2001 Thorup and Zwick devised a distance oracle, which given an $n$-vertex undirected graph and a parameter $k$, has size $O(k n^{1+1/k})$. Upon a query $(u,v)$ their oracle constructs a $(2k-1)$-approximate path $\Pi$ between $u$ and $v$. The query time of the Thorup-Zwick's oracle is $O(k)$, and it was subsequently improved to $O(1)$ by Chechik. A major drawback of the oracle of Thorup and Zwick is that its space is $\Omega(n \cdot \log n)$. Mendel and Naor devised an oracle with space $O(n^{1+1/k})$ and stretch $O(k)$, but their oracle can only report distance estimates and not actual paths. In this paper we devise a path-reporting distance oracle with size $O(n^{1+1/k})$, stretch $O(k)$ and query time $O(n^\epsilon)$, for an arbitrarily small $\epsilon > 0$. In particular, our oracle can provide logarithmic stretch using linear size. Another variant of our oracle has size $O(n \log\log n)$, polylogarithmic stretch, and query time $O(\log\log n)$. For unweighted graphs we devise a distance oracle with multiplicative stretch $O(1)$, additive stretch $O(\beta(k))$, for a function $\beta(\cdot)$, space $O(n^{1+1/k} \cdot \beta)$, and query time $O(n^\epsilon)$, for an arbitrarily small constant $\epsilon >0$. The tradeoff between multiplicative stretch and size in these oracles is far below girth conjecture threshold (which is stretch $2k-1$ and size $O(n^{1+1/k})$). Breaking the girth conjecture tradeoff is achieved by exhibiting a tradeoff of different nature between additive stretch $\beta(k)$ and size $O(n^{1+1/k})$. A similar type of tradeoff was exhibited by a construction of $(1+\epsilon,\beta)$-spanners due to Elkin and Peleg. However, so far $(1+\epsilon,\beta)$-spanners had no counterpart in the distance oracles' world. An important novel tool that we develop on the way to these results is a {distance-preserving path-reporting oracle}

Parallel Algorithms for Geometric Graph Problems

We give algorithms for geometric graph problems in the modern parallel models inspired by MapReduce. For example, for the Minimum Spanning Tree (MST) problem over a set of points in the two-dimensional space, our algorithm computes a $(1+\epsilon)$-approximate MST. Our algorithms work in a constant number of rounds of communication, while using total space and communication proportional to the size of the data (linear space and near linear time algorithms). In contrast, for general graphs, achieving the same result for MST (or even connectivity) remains a challenging open problem, despite drawing significant attention in recent years. We develop a general algorithmic framework that, besides MST, also applies to Earth-Mover Distance (EMD) and the transportation cost problem. Our algorithmic framework has implications beyond the MapReduce model. For example it yields a new algorithm for computing EMD cost in the plane in near-linear time, $n^{1+o_\epsilon(1)}$. We note that while recently Sharathkumar and Agarwal developed a near-linear time algorithm for $(1+\epsilon)$-approximating EMD, our algorithm is fundamentally different, and, for example, also solves the transportation (cost) problem, raised as an open question in their work. Furthermore, our algorithm immediately gives a $(1+\epsilon)$-approximation algorithm with $n^{\delta}$ space in the streaming-with-sorting model with $1/\delta^{O(1)}$ passes. As such, it is tempting to conjecture that the parallel models may also constitute a concrete playground in the quest for efficient algorithms for EMD (and other similar problems) in the vanilla streaming model, a well-known open problem

Bayesian Model Averaging in the Context of Spatial Hedonic Pricing: An Application to Farmland Values

Since 1973, British Columbia created an Agricultural Land Reserve to protect farmland from development. In this study, we employ GIS-based hedonic pricing models of farmland values to examine factors that affect farmland prices. We take spatial lag and error dependence into explicit account. However, the use of spatial econometric techniques in hedonic pricing models is problematic because there is uncertainty with respect to the choice of the explanatory variables and the spatial weighting matrix. Bayesian model averaging techniques in combination with Markov Chain Monte Carlo Model Composition are used to allow for both types of model uncertainty.Bayesian model averaging, Markov Chain Monte Carlo Model Composition, spatial econometrics, hedonic pricing, GIS, urban-rural fringe, farmland fragmentation

Learning Representations from EEG with Deep Recurrent-Convolutional Neural Networks

One of the challenges in modeling cognitive events from electroencephalogram (EEG) data is finding representations that are invariant to inter- and intra-subject differences, as well as to inherent noise associated with such data. Herein, we propose a novel approach for learning such representations from multi-channel EEG time-series, and demonstrate its advantages in the context of mental load classification task. First, we transform EEG activities into a sequence of topology-preserving multi-spectral images, as opposed to standard EEG analysis techniques that ignore such spatial information. Next, we train a deep recurrent-convolutional network inspired by state-of-the-art video classification to learn robust representations from the sequence of images. The proposed approach is designed to preserve the spatial, spectral, and temporal structure of EEG which leads to finding features that are less sensitive to variations and distortions within each dimension. Empirical evaluation on the cognitive load classification task demonstrated significant improvements in classification accuracy over current state-of-the-art approaches in this field.Comment: To be published as a conference paper at ICLR 201

Secluded Connectivity Problems

Consider a setting where possibly sensitive information sent over a path in a network is visible to every {neighbor} of the path, i.e., every neighbor of some node on the path, thus including the nodes on the path itself. The exposure of a path $P$ can be measured as the number of nodes adjacent to it, denoted by $N[P]$. A path is said to be secluded if its exposure is small. A similar measure can be applied to other connected subgraphs, such as Steiner trees connecting a given set of terminals. Such subgraphs may be relevant due to considerations of privacy, security or revenue maximization. This paper considers problems related to minimum exposure connectivity structures such as paths and Steiner trees. It is shown that on unweighted undirected $n$-node graphs, the problem of finding the minimum exposure path connecting a given pair of vertices is strongly inapproximable, i.e., hard to approximate within a factor of $O(2^{\log^{1-\epsilon}n})$ for any $\epsilon>0$ (under an appropriate complexity assumption), but is approximable with ratio $\sqrt{\Delta}+3$, where $\Delta$ is the maximum degree in the graph. One of our main results concerns the class of bounded-degree graphs, which is shown to exhibit the following interesting dichotomy. On the one hand, the minimum exposure path problem is NP-hard on node-weighted or directed bounded-degree graphs (even when the maximum degree is 4). On the other hand, we present a polynomial algorithm (based on a nontrivial dynamic program) for the problem on unweighted undirected bounded-degree graphs. Likewise, the problem is shown to be polynomial also for the class of (weighted or unweighted) bounded-treewidth graphs