122 research outputs found
Lower bounds on the dilation of plane spanners
(I) We exhibit a set of 23 points in the plane that has dilation at least
, improving the previously best lower bound of for the
worst-case dilation of plane spanners.
(II) For every integer , there exists an -element point set
such that the degree 3 dilation of denoted by in the domain of plane geometric spanners. In the
same domain, we show that for every integer , there exists a an
-element point set such that the degree 4 dilation of denoted by
The
previous best lower bound of holds for any degree.
(III) For every integer , there exists an -element point set
such that the stretch factor of the greedy triangulation of is at least
.Comment: Revised definitions in the introduction; 23 pages, 15 figures; 2
table
Computing the Greedy Spanner in Linear Space
The greedy spanner is a high-quality spanner: its total weight, edge count
and maximal degree are asymptotically optimal and in practice significantly
better than for any other spanner with reasonable construction time.
Unfortunately, all known algorithms that compute the greedy spanner of n points
use Omega(n^2) space, which is impractical on large instances. To the best of
our knowledge, the largest instance for which the greedy spanner was computed
so far has about 13,000 vertices.
We present a O(n)-space algorithm that computes the same spanner for points
in R^d running in O(n^2 log^2 n) time for any fixed stretch factor and
dimension. We discuss and evaluate a number of optimizations to its running
time, which allowed us to compute the greedy spanner on a graph with a million
vertices. To our knowledge, this is also the first algorithm for the greedy
spanner with a near-quadratic running time guarantee that has actually been
implemented
Algorithmic and Combinatorial Results on Fence Patrolling, Polygon Cutting and Geometric Spanners
The purpose of this dissertation is to study problems that lie at the intersection of geometry and computer science. We have studied and obtained several results from three different areas, namely–geometric spanners, polygon cutting, and fence patrolling. Specifically, we have designed and analyzed algorithms along with various combinatorial results in these three areas. For geometric spanners, we have obtained combinatorial results regarding lower bounds on worst case dilation of plane spanners. We also have studied low degree plane lattice spanners, both square and hexagonal, of low dilation. Next, for polygon cutting, we have designed and analyzed algorithms for cutting out polygon collections drawn on a piece of planar material
using the three geometric models of saw, namely, line, ray and segment cuts. For fence patrolling, we have designed several strategies for robots patrolling both open and closed fences
Feed-links for network extensions
Road network data is often incomplete, making it hard to perform network analysis. This paper discusses the problem of extending partial road networks with reasonable links, using the concept of dilation (also known as crow flight conversion coefficient). To this end, we study how to connect a point (relevant location) inside a polygon (face of the known part of the road network) to the boundary so that the dilation from that point to any point on the boundary is not too large. We provide algorithms and heuristics, and give a computational and experimental analysis
Improving the dilation of a metric graph by adding edges
Most of the literature on spanners focuses on building the graph from
scratch. This paper instead focuses on adding edges to improve an existing
graph. A major open problem in this field is: given a graph embedded in a
metric space, and a budget of k edges, which k edges do we add to produce a
minimum-dilation graph? The special case where k=1 has been studied in the
past, but no major breakthroughs have been made for k > 1. We provide the first
positive result, an O(k)-approximation algorithm that runs in O(n^3 \log n)
time
Locating Battery Charging Stations to Facilitate Almost Shortest Paths
We study a facility location problem motivated by requirements pertaining to the distribution of charging stations for electric vehicles: Place a minimum number of battery charging stations at a subset of nodes of a network, so that battery-powered electric vehicles will be able to move between destinations using "t-spanning" routes, of lengths within a factor t > 1 of the length of a shortest path, while having sufficient charging stations along the way. We give constant-factor approximation algorithms for minimizing the number of charging stations, subject to the t-spanning constraint. We study two versions of the problem, one in which the stations are required to support a single ride (to a single destination), and one in which the stations are to support multiple rides through a sequence of destinations, where the destinations are revealed one at a time
Linear Time Algorithm for Optimal Feed-link Placement
Given a polygon representing a transportation network together with a point p
in its interior, we aim to extend the network by inserting a line segment,
called a feed-link, which connects p to the boundary of the polygon. Once a
feed link is fixed, the geometric dilation of some point q on the boundary is
the ratio between the length of the shortest path from p to q through the
extended network, and their Euclidean distance. The utility of a feed-link is
inversely proportional to the maximal dilation over all boundary points.
We give a linear time algorithm for computing the feed-link with the minimum
overall dilation, thus improving upon the previously known algorithm of
complexity that is roughly O(n log n)
On the relativistic viability of multi-automaton systems: essential concepts, challenges and prospects
Our understanding of the Universe breaks down for very small spacetime
intervals, corresponding to an extremely high level of granularity (and
energy), commonly referred to as the ``Planck scale''. At this fundamental
level, there are attempts of describing physics in terms of interacting
automata that perform classical, deterministic computation. On one hand,
various mathematical arguments have already illustrated how quantum laws (which
describe elementary particles and interactions) could in principle arise as
low-granularity approximations of automata-based systems. On the other hand,
understanding how such systems might give rise to relativistic laws (which
describe spacetime and gravity) remains a major problem. I explain here a few
ideas that seem crucial for overcoming this problem, along with related
algorithmic challenges that need to be addressed. Giving emphasis to meaningful
computational counterparts of locality and general covariance, I outline basic
ingredients of a distributed communication-rewiring protocol that would allow
us to construct multi-automaton models that are viable from a relativistic
perspective. I also explain how viable models can be evaluated using a variety
of criteria, and discuss related aspects pertaining to the falsifiability and
plausibility of the automata paradigm.Comment: 7 pages, 1 figur
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