731 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 a family of strong geometric spanners that admit local routing strategies
We introduce a family of directed geometric graphs, denoted \paz, that
depend on two parameters and . For and , the \paz graph is a strong
-spanner, with . The out-degree of a node
in the \paz graph is at most . Moreover, we show that routing can be
achieved locally on \paz. Next, we show that all strong -spanners are also
-spanners of the unit disk graph. Simulations for various values of the
parameters and indicate that for random point sets, the
spanning ratio of \paz is better than the proven theoretical bounds
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
Sparse geometric graphs with small dilation
Given a set S of n points in R^D, and an integer k such that 0 <= k < n, we
show that a geometric graph with vertex set S, at most n - 1 + k edges, maximum
degree five, and dilation O(n / (k+1)) can be computed in time O(n log n). For
any k, we also construct planar n-point sets for which any geometric graph with
n-1+k edges has dilation Omega(n/(k+1)); a slightly weaker statement holds if
the points of S are required to be in convex position
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
Spanning Properties of Theta-Theta Graphs
We study the spanning properties of Theta-Theta graphs. Similar in spirit
with the Yao-Yao graphs, Theta-Theta graphs partition the space around each
vertex into a set of k cones, for some fixed integer k > 1, and select at most
one edge per cone. The difference is in the way edges are selected. Yao-Yao
graphs select an edge of minimum length, whereas Theta-Theta graphs select an
edge of minimum orthogonal projection onto the cone bisector. It has been
established that the Yao-Yao graphs with parameter k = 6k' have spanning ratio
11.67, for k' >= 6. In this paper we establish a first spanning ratio of
for Theta-Theta graphs, for the same values of . We also extend the class of
Theta-Theta spanners with parameter 6k', and establish a spanning ratio of
for k' >= 5. We surmise that these stronger results are mainly due to a
tighter analysis in this paper, rather than Theta-Theta being superior to
Yao-Yao as a spanner. We also show that the spanning ratio of Theta-Theta
graphs decreases to 4.64 as k' increases to 8. These are the first results on
the spanning properties of Theta-Theta graphs.Comment: 20 pages, 6 figures, 3 table
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