37 research outputs found
Geometric Spanners for Points Inside a Polygonal Domain
Let P be a set of n points inside a polygonal domain D. A polygonal domain with h holes (or obstacles) consists of h disjoint polygonal obstacles surrounded by a simple polygon which itself acts as an obstacle. We first study t-spanners for the set P with respect to the geodesic distance function d where for any two points p and q, d(p,q) is equal to the Euclidean length of the shortest path from p to q that avoids the obstacles interiors. For a case where the polygonal domain is a simple polygon (i.e., h=0), we construct a (sqrt(10)+eps)-spanner that has O(n log^2 n) edges where eps is the a given positive real number. For a case where there are h holes, our construction gives a (5+eps)-spanner with the size of O(sqrt(h) n log^2 n).
Moreover, we study t-spanners for the visibility graph of P (VG(P), for short) with respect to a hole-free polygonal domain D. The graph VG(P) is not necessarily a complete graph or even connected. In this case, we propose an algorithm that constructs a (3+eps)-spanner of size almost O(n^{4/3}). In addition, we show that there is a set P of n points such that any (3-eps)-spanner of VG(P) must contain almost n^2 edges
On the power of the semi-separated pair decomposition
A Semi-Separated Pair Decomposition (SSPD), with parameter s > 1, of a set is a set {(A i ,B i )} of pairs of subsets of S such that for each i, there are balls and containing A i and B i respectively such that min ( radius ) , radius ), and for any two points p, q S there is a unique index i such that p A i and q B i or vice-versa. In this paper, we use the SSPD to obtain the following results: First, we consider the construction of geometric t-spanners in the context of imprecise points and we prove that any set of n imprecise points, modeled as pairwise disjoint balls, admits a t-spanner with edges which can be computed in time. If all balls have the same radius, the number of edges reduces to . Secondly, for a set of n points in the plane, we design a query data structure for half-plane closest-pair queries that can be built in time using space and answers a query in time, for any ε> 0. By reducing the preprocessing time to and using space, the query can be answered in time. Moreover, we improve the preprocessing time of an existing axis-parallel rectangle closest-pair query data structure from quadratic to near-linear. Finally, we revisit some previously studied problems, namely spanners for complete k-partite graphs and l
Convex Hull of Points Lying on Lines in o(n log n) Time after Preprocessing
Motivated by the desire to cope with data imprecision, we study methods for
taking advantage of preliminary information about point sets in order to speed
up the computation of certain structures associated with them.
In particular, we study the following problem: given a set L of n lines in
the plane, we wish to preprocess L such that later, upon receiving a set P of n
points, each of which lies on a distinct line of L, we can construct the convex
hull of P efficiently. We show that in quadratic time and space it is possible
to construct a data structure on L that enables us to compute the convex hull
of any such point set P in O(n alpha(n) log* n) expected time. If we further
assume that the points are "oblivious" with respect to the data structure, the
running time improves to O(n alpha(n)). The analysis applies almost verbatim
when L is a set of line-segments, and yields similar asymptotic bounds. We
present several extensions, including a trade-off between space and query time
and an output-sensitive algorithm. We also study the "dual problem" where we
show how to efficiently compute the (<= k)-level of n lines in the plane, each
of which lies on a distinct point (given in advance).
We complement our results by Omega(n log n) lower bounds under the algebraic
computation tree model for several related problems, including sorting a set of
points (according to, say, their x-order), each of which lies on a given line
known in advance. Therefore, the convex hull problem under our setting is
easier than sorting, contrary to the "standard" convex hull and sorting
problems, in which the two problems require Theta(n log n) steps in the worst
case (under the algebraic computation tree model).Comment: 26 pages, 5 figures, 1 appendix; a preliminary version appeared at
SoCG 201