48,101 research outputs found

    Cover Time and Broadcast Time

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    We introduce a new technique for bounding the cover time of random walks by relating it to the runtime of randomized broadcast. In particular, we strongly confirm for dense graphs the intuition of Chandra et al. (1997) that ``the cover time of the graph is an appropriate metric for the performance of certain kinds of randomized broadcast algorithms\u27\u27. In more detail, our results are as follows: begin{itemize} item For any graph G=(V,E)G=(V,E) of size nn and minimum degree deltadelta, we have mathcalR(G)=mathcalO(frac∣E∣deltacdotlogn)mathcal{R}(G)= mathcal{O}(frac{|E|}{delta} cdot log n), where mathcalR(G)mathcal{R}(G) denotes the quotient of the cover time and broadcast time. This bound is tight for binary trees and tight up to logarithmic factors for many graphs including hypercubes, expanders and lollipop graphs. item For any deltadelta-regular (or almost deltadelta-regular) graph GG it holds that mathcalR(G)=Omega(fracdelta2ncdotfrac1logn)mathcal{R}(G) = Omega(frac{delta^2}{n} cdot frac{1}{log n}). Together with our upper bound on mathcalR(G)mathcal{R}(G), this lower bound strongly confirms the intuition of Chandra et al.~for graphs with minimum degree Theta(n)Theta(n), since then the cover time equals the broadcast time multiplied by nn (neglecting logarithmic factors). item Conversely, for any deltadelta we construct almost deltadelta-regular graphs that satisfy mathcalR(G)=mathcalO(maxsqrtn,deltacdotlog2n)mathcal{R}(G) = mathcal{O}(max { sqrt{n},delta } cdot log^2 n). Since any regular expander satisfies mathcalR(G)=Theta(n)mathcal{R}(G) = Theta(n), the strong relationship given above does not hold if deltadelta is polynomially smaller than nn. end{itemize} Our bounds also demonstrate that the relationship between cover time and broadcast time is much stronger than the known relationships between any of them and the mixing time (or the closely related spectral gap)

    The Quadratic Cycle Cover Problem: special cases and efficient bounds

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    The quadratic cycle cover problem is the problem of finding a set of node-disjoint cycles visiting all the nodes such that the total sum of interaction costs between consecutive arcs is minimized. In this paper we study the linearization problem for the quadratic cycle cover problem and related lower bounds. In particular, we derive various sufficient conditions for the quadratic cost matrix to be linearizable, and use these conditions to compute bounds. We also show how to use a sufficient condition for linearizability within an iterative bounding procedure. In each step, our algorithm computes the best equivalent representation of the quadratic cost matrix and its optimal linearizable matrix with respect to the given sufficient condition for linearizability. Further, we show that the classical Gilmore-Lawler type bound belongs to the family of linearization based bounds, and therefore apply the above mentioned iterative reformulation technique. We also prove that the linearization vectors resulting from this iterative approach satisfy the constant value property. The best among here introduced bounds outperform existing lower bounds when taking both quality and efficiency into account

    A Novel Approach for Ellipsoidal Outer-Approximation of the Intersection Region of Ellipses in the Plane

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    In this paper, a novel technique for tight outer-approximation of the intersection region of a finite number of ellipses in 2-dimensional (2D) space is proposed. First, the vertices of a tight polygon that contains the convex intersection of the ellipses are found in an efficient manner. To do so, the intersection points of the ellipses that fall on the boundary of the intersection region are determined, and a set of points is generated on the elliptic arcs connecting every two neighbouring intersection points. By finding the tangent lines to the ellipses at the extended set of points, a set of half-planes is obtained, whose intersection forms a polygon. To find the polygon more efficiently, the points are given an order and the intersection of the half-planes corresponding to every two neighbouring points is calculated. If the polygon is convex and bounded, these calculated points together with the initially obtained intersection points will form its vertices. If the polygon is non-convex or unbounded, we can detect this situation and then generate additional discrete points only on the elliptical arc segment causing the issue, and restart the algorithm to obtain a bounded and convex polygon. Finally, the smallest area ellipse that contains the vertices of the polygon is obtained by solving a convex optimization problem. Through numerical experiments, it is illustrated that the proposed technique returns a tighter outer-approximation of the intersection of multiple ellipses, compared to conventional techniques, with only slightly higher computational cost

    The performance of object decomposition techniques for spatial query processing

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    Focusing on the Big Picture: Insights into a Systems Approach to Deep Learning for Satellite Imagery

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    Deep learning tasks are often complicated and require a variety of components working together efficiently to perform well. Due to the often large scale of these tasks, there is a necessity to iterate quickly in order to attempt a variety of methods and to find and fix bugs. While participating in IARPA's Functional Map of the World challenge, we identified challenges along the entire deep learning pipeline and found various solutions to these challenges. In this paper, we present the performance, engineering, and deep learning considerations with processing and modeling data, as well as underlying infrastructure considerations that support large-scale deep learning tasks. We also discuss insights and observations with regard to satellite imagery and deep learning for image classification.Comment: Accepted to IEEE Big Data 201
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