365 research outputs found
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
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
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