341 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
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
Balancing Degree, Diameter and Weight in Euclidean Spanners
In this paper we devise a novel \emph{unified} construction of Euclidean
spanners that trades between the maximum degree, diameter and weight
gracefully. For a positive integer k, our construction provides a
(1+eps)-spanner with maximum degree O(k), diameter O(log_k n + alpha(k)),
weight O(k \cdot log_k n \cdot log n) \cdot w(MST(S)), and O(n) edges. Note
that for k= n^{1/alpha(n)} this gives rise to diameter O(alpha(n)), weight
O(n^{1/alpha(n)} \cdot log n \cdot alpha(n)) \cdot w(MST(S)) and maximum degree
O(n^{1/alpha(n)}), which improves upon a classical result of Arya et al.
\cite{ADMSS95}; in the corresponding result from \cite{ADMSS95} the spanner has
the same number of edges and diameter, but its weight and degree may be
arbitrarily large. Also, for k = O(1) this gives rise to maximum degree O(1),
diameter O(log n) and weight O(log^2 n) \cdot w(MST(S)), which reproves another
classical result of Arya et al. \cite{ADMSS95}. Our bound of O(log_k n +
alpha(k)) on the diameter is optimal under the constraints that the maximum
degree is O(k) and the number of edges is O(n). Our bound on the weight is
optimal up to a factor of log n. Our construction also provides a similar
tradeoff in the complementary range of parameters, i.e., when the weight should
be smaller than log^2 n, but the diameter is allowed to grow beyond log n.
For random point sets in the d-dimensional unit cube, we "shave" a factor of
log n from the weight bound. Specifically, in this case our construction
achieves maximum degree O(k), diameter O(log_k n + alpha(k)) and weight that is
with high probability O(k \cdot log_k n) \cdot w(MST(S)).
Finally, en route to these results we devise optimal constructions of
1-spanners for general tree metrics, which are of independent interest.Comment: 27 pages, 7 figures; a preliminary version of this paper appeared in
ESA'1
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
Massively Parallel Algorithms for Distance Approximation and Spanners
Over the past decade, there has been increasing interest in
distributed/parallel algorithms for processing large-scale graphs. By now, we
have quite fast algorithms -- usually sublogarithmic-time and often
-time, or even faster -- for a number of fundamental graph
problems in the massively parallel computation (MPC) model. This model is a
widely-adopted theoretical abstraction of MapReduce style settings, where a
number of machines communicate in an all-to-all manner to process large-scale
data. Contributing to this line of work on MPC graph algorithms, we present
round MPC algorithms for computing
-spanners in the strongly sublinear regime of local memory. To
the best of our knowledge, these are the first sublogarithmic-time MPC
algorithms for spanner construction. As primary applications of our spanners,
we get two important implications, as follows:
-For the MPC setting, we get an -round algorithm for
approximation of all pairs shortest paths (APSP) in the
near-linear regime of local memory. To the best of our knowledge, this is the
first sublogarithmic-time MPC algorithm for distance approximations.
-Our result above also extends to the Congested Clique model of distributed
computing, with the same round complexity and approximation guarantee. This
gives the first sub-logarithmic algorithm for approximating APSP in weighted
graphs in the Congested Clique model
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