48,101 research outputs found
Cover Time and Broadcast Time
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 of size and minimum degree , we have , where 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 -regular (or almost -regular) graph it holds that . Together with our upper bound on , this lower bound strongly confirms the intuition of Chandra et al.~for graphs with minimum degree , since then the cover time equals the broadcast time multiplied by (neglecting logarithmic factors).
item Conversely, for any we construct almost -regular graphs that satisfy . Since any regular expander satisfies , the strong relationship given above does not hold if is polynomially smaller than .
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
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
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
Focusing on the Big Picture: Insights into a Systems Approach to Deep Learning for Satellite Imagery
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
- …