52 research outputs found
Two Approaches to Building Time-Windowed Geometric Data Structures
Given a set of geometric objects each associated with a time value, we wish to determine whether a given property is true for a subset of those objects whose time values fall within a query time window. We call such problems time-windowed decision problems, and they have been the subject of much recent attention, for instance studied by Bokal, Cabello, and Eppstein [SoCG 2015]. In this paper, we present new approaches to this class of problems that are conceptually simpler than Bokal et al.\u27s, and also lead to faster algorithms. For instance, we present algorithms for preprocessing for the time-windowed 2D diameter decision problem in O(n log n) time and the time-windowed 2D convex hull area decision problem in O(n alpha(n) log n) time (where alpha is the inverse Ackermann function), improving Bokal et al.\u27s O(n log^2 n) and O(n log n loglog n) solutions respectively.
Our first approach is to reduce time-windowed decision problems to a generalized range successor problem, which we solve using a novel way to search range trees. Our other approach is to use dynamic data structures directly, taking advantage of a new observation that the total number of combinatorial changes to a planar convex hull is near linear for any FIFO update sequence, in which deletions occur in the same order as insertions. We also apply these approaches to obtain the first O(n polylog n) algorithms for the time-windowed 3D diameter decision and 2D orthogonal segment intersection detection problems
On the Extremal Functions of Acyclic Forbidden 0--1 Matrices
The extremal theory of forbidden 0--1 matrices studies the asymptotic growth
of the function , which is the maximum weight of a matrix
whose submatrices avoid a fixed pattern
. This theory has been wildly successful at resolving
problems in combinatorics, discrete and computational geometry, structural
graph theory, and the analysis of data structures, particularly corollaries of
the dynamic optimality conjecture.
All these applications use acyclic patterns, meaning that when is
regarded as the adjacency matrix of a bipartite graph, the graph is acyclic.
The biggest open problem in this area is to bound for
acyclic . Prior results have only ruled out the strict bound
conjectured by Furedi and Hajnal. It is consistent with prior results that
, and also consistent that
.
In this paper we establish a stronger lower bound on the extremal functions
of acyclic . Specifically, we give a new construction of relatively dense
0--1 matrices with 1s that avoid an acyclic
. Pach and Tardos have conjectured that this type of result is the best
possible, i.e., no acyclic exists for which
Largest Similar Copies of Convex Polygons in Polygonal Domains
Given a convex polygon with k vertices and a polygonal domain consisting of polygonal obstacles with n vertices in total in the plane, we study the optimization problem of finding a largest similar copy of the polygon that can be placed in the polygonal domain without intersecting the obstacles. We present an upper bound O(k1
Computational Geometry Column 34
Problems presented at the open-problem session of the 14th Annual ACM
Symposium on Computational Geometry are listed
On the Complexity of Randomly Weighted Voronoi Diagrams
In this paper, we provide an bound on the expected
complexity of the randomly weighted Voronoi diagram of a set of sites in
the plane, where the sites can be either points, interior-disjoint convex sets,
or other more general objects. Here the randomness is on the weight of the
sites, not their location. This compares favorably with the worst case
complexity of these diagrams, which is quadratic. As a consequence we get an
alternative proof to that of Agarwal etal [AHKS13] of the near linear
complexity of the union of randomly expanded disjoint segments or convex sets
(with an improved bound on the latter). The technique we develop is elegant and
should be applicable to other problems
Non-uniform Geometric Set Cover and Scheduling on Multiple Machines
We consider the following general scheduling problem studied recently by
Moseley. There are jobs, all released at time , where job has size
and an associated arbitrary non-decreasing cost function of its
completion time. The goal is to find a schedule on machines with minimum
total cost. We give an approximation for the problem, improving upon the
previous bound ( is the maximum to minimum size ratio),
and resolving the open question of Moseley.
We first note that the scheduling problem can be reduced to a clean geometric
set cover problem where points on a line with arbitrary demands, must be
covered by a minimum cost collection of given intervals with non-uniform
capacity profiles. Unfortunately, current techniques for such problems based on
knapsack cover inequalities and low union complexity, completely lose the
geometric structure in the non-uniform capacity profiles and incur at least an
loss.
To this end, we consider general covering problems with non-uniform
capacities, and give a new method to handle capacities in a way that completely
preserves their geometric structure. This allows us to use sophisticated
geometric ideas in a black-box way to avoid the loss in
previous approaches. In addition to the scheduling problem above, we use this
approach to obtain or inverse Ackermann type bounds for several basic
capacitated covering problems
Network Farthest-Point Diagrams
Consider the continuum of points along the edges of a network, i.e., an
undirected graph with positive edge weights. We measure distance between these
points in terms of the shortest path distance along the network, known as the
network distance. Within this metric space, we study farthest points.
We introduce network farthest-point diagrams, which capture how the farthest
points---and the distance to them---change as we traverse the network. We
preprocess a network G such that, when given a query point q on G, we can
quickly determine the farthest point(s) from q in G as well as the farthest
distance from q in G. Furthermore, we introduce a data structure supporting
queries for the parts of the network that are farther away from q than some
threshold R > 0, where R is part of the query.
We also introduce the minimum eccentricity feed-link problem defined as
follows. Given a network G with geometric edge weights and a point p that is
not on G, connect p to a point q on G with a straight line segment pq, called a
feed-link, such that the largest network distance from p to any point in the
resulting network is minimized. We solve the minimum eccentricity feed-link
problem using eccentricity diagrams. In addition, we provide a data structure
for the query version, where the network G is fixed and a query consists of the
point p.Comment: A preliminary version of this work was presented at the 24th Canadian
Conference on Computational Geometr
Kinetic Voronoi Diagrams and Delaunay Triangulations under Polygonal Distance Functions
Let be a set of points and a convex -gon in .
We analyze in detail the topological (or discrete) changes in the structure of
the Voronoi diagram and the Delaunay triangulation of , under the convex
distance function defined by , as the points of move along prespecified
continuous trajectories. Assuming that each point of moves along an
algebraic trajectory of bounded degree, we establish an upper bound of
on the number of topological changes experienced by the
diagrams throughout the motion; here is the maximum length of an
-Davenport-Schinzel sequence, and is a constant depending on the
algebraic degree of the motion of the points. Finally, we describe an algorithm
for efficiently maintaining the above structures, using the kinetic data
structure (KDS) framework
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