1,197 research outputs found
Shortcut sets for plane Euclidean networks (Extended abstract)
We study the problem of augmenting the locus N of a plane Euclidean network N
by inserting iteratively a finite set of segments, called shortcut set, while reducing
the diameter of the locus of the resulting network. We first characterize the existence
of shortcut sets, and compute shortcut sets in polynomial time providing an upper
bound on their size. Then, we analyze the role of the convex hull of N when
inserting a shortcut set. As a main result, we prove that one can always determine in
polynomial time whether inserting only one segment suffices to reduce the diameter.Ministerio de EconomĂa y Competitividad MTM2014-60127-PJunta de AndalucĂa FQM-016
Computing optimal shortcuts for networks
We augment a plane Euclidean network with a segment or shortcut to minimize the largest distance between any two points along the edges of the resulting network. In this continuous setting, the problem of computing distances and placing a shortcut is much harder as all points on the network, instead of only the vertices, must be taken into account. Our main result for general networks states that it is always possible to determine in polynomial time whether the network has an optimal shortcut and compute one in case of existence. We also improve this general method for networks that are paths, restricted to using two types of shortcuts: those of any fixed direction and shortcuts that intersect the path only on its endpoints.Peer ReviewedPostprint (published version
There are Plane Spanners of Maximum Degree 4
Let E be the complete Euclidean graph on a set of points embedded in the
plane. Given a constant t >= 1, a spanning subgraph G of E is said to be a
t-spanner, or simply a spanner, if for any pair of vertices u,v in E the
distance between u and v in G is at most t times their distance in E. A spanner
is plane if its edges do not cross.
This paper considers the question: "What is the smallest maximum degree that
can always be achieved for a plane spanner of E?" Without the planarity
constraint, it is known that the answer is 3 which is thus the best known lower
bound on the degree of any plane spanner. With the planarity requirement, the
best known upper bound on the maximum degree is 6, the last in a long sequence
of results improving the upper bound. In this paper we show that the complete
Euclidean graph always contains a plane spanner of maximum degree at most 4 and
make a big step toward closing the question. Our construction leads to an
efficient algorithm for obtaining the spanner from Chew's L1-Delaunay
triangulation
Spanners for Geometric Intersection Graphs
Efficient algorithms are presented for constructing spanners in geometric
intersection graphs. For a unit ball graph in R^k, a (1+\epsilon)-spanner is
obtained using efficient partitioning of the space into hypercubes and solving
bichromatic closest pair problems. The spanner construction has almost
equivalent complexity to the construction of Euclidean minimum spanning trees.
The results are extended to arbitrary ball graphs with a sub-quadratic running
time.
For unit ball graphs, the spanners have a small separator decomposition which
can be used to obtain efficient algorithms for approximating proximity problems
like diameter and distance queries. The results on compressed quadtrees,
geometric graph separators, and diameter approximation might be of independent
interest.Comment: 16 pages, 5 figures, Late
Semantic Interpretation of an Artificial Neural Network
Recent advances in machine learning theory have opened the door for applications to many difficult problem domains. One area that has achieved great success for stock market analysis/prediction is artificial neural networks. However, knowledge embedded in the neural network is not easily translated into symbolic form. Recent research, exploring the viability of merging artificial neural networks with traditional rule-based expert systems, has achieved limited success. In particular, extracting production (IF.. THEN) rules from a trained neural net based on connection weights provides a valid set of rules only when neuron outputs are close to 0 or 1 (e.g. the output sigmoid function is saturated). This thesis presents two new ways to interpret neural network knowledge. The first, called Knowledge Math, extends the use of connection weights, generating rules for general (i.e. non-binary) input and output values. The second method, based on decision boundaries, utilizes the inherent border between output classification regions to draw symbolic interpretation. The Decision Boundary method generates more complex symbolic rules than Knowledge Math, but provides valid feature relationships in the uncertain regions around the midpoints of the neuron output functions. The main result is a complementary relationship between Knowledge Math and Decision Boundaries, as well as subsymbolic and symbolic knowledge representations for a general multi-layer perceptron
Scalable Routing Easy as PIE: a Practical Isometric Embedding Protocol (Technical Report)
We present PIE, a scalable routing scheme that achieves 100% packet delivery
and low path stretch. It is easy to implement in a distributed fashion and
works well when costs are associated to links. Scalability is achieved by using
virtual coordinates in a space of concise dimensionality, which enables greedy
routing based only on local knowledge. PIE is a general routing scheme, meaning
that it works on any graph. We focus however on the Internet, where routing
scalability is an urgent concern. We show analytically and by using simulation
that the scheme scales extremely well on Internet-like graphs. In addition, its
geometric nature allows it to react efficiently to topological changes or
failures by finding new paths in the network at no cost, yielding better
delivery ratios than standard algorithms. The proposed routing scheme needs an
amount of memory polylogarithmic in the size of the network and requires only
local communication between the nodes. Although each node constructs its
coordinates and routes packets locally, the path stretch remains extremely low,
even lower than for centralized or less scalable state-of-the-art algorithms:
PIE always finds short paths and often enough finds the shortest paths.Comment: This work has been previously published in IEEE ICNP'11. The present
document contains an additional optional mechanism, presented in Section
III-D, to further improve performance by using route asymmetry. It also
contains new simulation result
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