2,254 research outputs found
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
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
Remote spanners: what to know beyond neighbors
International audienceMotivated by the fact that neighbors are generally known in practical routing algorithms, we introduce the notion of remote-spanner. Given an unweighted graph , a sub-graph with vertex set is an \emph{(\a,\b)-remote-spanner} if for each pair of points and the distance between and in , the graph augmented by all the edges between and its neighbors in , is at most \a times the distance between and in plus \b. We extend this definition to -connected graphs by considering minimum length sum over disjoint paths as distance. We then say that an (\a,\b)-remote-spanner is \emph{-connecting }. In this paper, we give distributed algorithms for computing (1+\eps,1-2\eps)-remote-spanners for any \eps>0, -connecting -remote-spanners for any (yielding -remote-spanners for ) and -connecting -remote-spanners. All these algorithms run in constant time for any unweighted input graph. The number of edges obtained for -connecting -remote-spanner is within a logarithmic factor from optimal (compared to the best -connecting -remote-spanner of the input graph). Interestingly, sparse -remote-spanners (i.e. preserving exact distances) with edges exist in random unit disk graphs. The number of edges obtained for (1+\eps,1-2\eps)-remote-spanners and -connecting -remote-spanners is linear if the input graph is the unit ball graph of a doubling metric (distances between nodes are unknown). Our methodology consists in characterizing remote-spanners as sub-graphs containing the union of small depth tree sub-graphs dominating nearby nodes. This leads to simple local distributed algorithms
Fractal Dimension and Lower Bounds for Geometric Problems
We study the complexity of geometric problems on spaces of low fractal dimension. It was recently shown by [Sidiropoulos & Sridhar, SoCG 2017] that several problems admit improved solutions when the input is a pointset in Euclidean space with fractal dimension smaller than the ambient dimension. In this paper we prove nearly-matching lower bounds, thus establishing nearly-optimal bounds for various problems as a function of the fractal dimension.
More specifically, we show that for any set of n points in d-dimensional Euclidean space, of fractal dimension delta in (1,d), for any epsilon>0 and c >= 1, any c-spanner must have treewidth at least Omega(n^{1-1/(delta - epsilon)} / c^{d-1}), matching the previous upper bound. The construction used to prove this lower bound on the treewidth of spanners, can also be used to derive lower bounds on the running time of algorithms for various problems, assuming the Exponential Time Hypothesis. We provide two prototypical results of this type:
- For any delta in (1,d) and any epsilon >0, d-dimensional Euclidean TSP on n points with fractal dimension at most delta cannot be solved in time 2^{O(n^{1-1/(delta - epsilon)})}. The best-known upper bound is 2^{O(n^{1-1/delta} log n)}.
- For any delta in (1,d) and any epsilon >0, the problem of finding k-pairwise non-intersecting d-dimensional unit balls/axis parallel unit cubes with centers having fractal dimension at most delta cannot be solved in time f(k)n^{O (k^{1-1/(delta - epsilon)})} for any computable function f. The best-known upper bound is n^{O(k^{1-1/delta} log n)}. The above results nearly match previously known upper bounds from [Sidiropoulos & Sridhar, SoCG 2017], and generalize analogous lower bounds for the case of ambient dimension due to [Marx & Sidiropoulos, SoCG 2014]
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