12 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
On Geometric Spanners of Euclidean and Unit Disk Graphs
We consider the problem of constructing bounded-degree planar geometric
spanners of Euclidean and unit-disk graphs. It is well known that the Delaunay
subgraph is a planar geometric spanner with stretch factor C_{del\approx
2.42; however, its degree may not be bounded. Our first result is a very
simple linear time algorithm for constructing a subgraph of the Delaunay graph
with stretch factor \rho =1+2\pi(k\cos{\frac{\pi{k)^{-1 and degree bounded by
, for any integer parameter . This result immediately implies an
algorithm for constructing a planar geometric spanner of a Euclidean graph with
stretch factor \rho \cdot C_{del and degree bounded by , for any integer
parameter . Moreover, the resulting spanner contains a Euclidean
Minimum Spanning Tree (EMST) as a subgraph. Our second contribution lies in
developing the structural results necessary to transfer our analysis and
algorithm from Euclidean graphs to unit disk graphs, the usual model for
wireless ad-hoc networks. We obtain a very simple distributed, {\em
strictly-localized algorithm that, given a unit disk graph embedded in the
plane, constructs a geometric spanner with the above stretch factor and degree
bound, and also containing an EMST as a subgraph. The obtained results
dramatically improve the previous results in all aspects, as shown in the
paper
Fault-tolerant additive weighted geometric spanners
Let S be a set of n points and let w be a function that assigns non-negative
weights to points in S. The additive weighted distance d_w(p, q) between two
points p,q belonging to S is defined as w(p) + d(p, q) + w(q) if p \ne q and it
is zero if p = q. Here, d(p, q) denotes the (geodesic) Euclidean distance
between p and q. A graph G(S, E) is called a t-spanner for the additive
weighted set S of points if for any two points p and q in S the distance
between p and q in graph G is at most t.d_w(p, q) for a real number t > 1.
Here, d_w(p,q) is the additive weighted distance between p and q. For some
integer k \geq 1, a t-spanner G for the set S is a (k, t)-vertex fault-tolerant
additive weighted spanner, denoted with (k, t)-VFTAWS, if for any set S'
\subset S with cardinality at most k, the graph G \ S' is a t-spanner for the
points in S \ S'. For any given real number \epsilon > 0, we obtain the
following results:
- When the points in S belong to Euclidean space R^d, an algorithm to compute
a (k,(2 + \epsilon))-VFTAWS with O(kn) edges for the metric space (S, d_w).
Here, for any two points p, q \in S, d(p, q) is the Euclidean distance between
p and q in R^d.
- When the points in S belong to a simple polygon P, for the metric space (S,
d_w), one algorithm to compute a geodesic (k, (2 + \epsilon))-VFTAWS with
O(\frac{k n}{\epsilon^{2}}\lg{n}) edges and another algorithm to compute a
geodesic (k, (\sqrt{10} + \epsilon))-VFTAWS with O(kn(\lg{n})^2) edges. Here,
for any two points p, q \in S, d(p, q) is the geodesic Euclidean distance along
the shortest path between p and q in P.
- When the points in lie on a terrain T, an algorithm to compute a
geodesic (k, (2 + \epsilon))-VFTAWS with O(\frac{k n}{\epsilon^{2}}\lg{n})
edges.Comment: a few update
Robust Geometric Spanners
Highly connected and yet sparse graphs (such as expanders or graphs of high
treewidth) are fundamental, widely applicable and extensively studied
combinatorial objects. We initiate the study of such highly connected graphs
that are, in addition, geometric spanners. We define a property of spanners
called robustness. Informally, when one removes a few vertices from a robust
spanner, this harms only a small number of other vertices. We show that robust
spanners must have a superlinear number of edges, even in one dimension. On the
positive side, we give constructions, for any dimension, of robust spanners
with a near-linear number of edges.Comment: 18 pages, 8 figure
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
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