835 research outputs found
Dynamic Connectivity in Disk Graphs
Let S ⊆ R2 be a set of n sites in the plane, so that every site s ∈ S has an associated
radius rs > 0. Let D(S) be the disk intersection graph defined by S, i.e., the graph
with vertex set S and an edge between two distinct sites s, t ∈ S if and only if the
disks with centers s, t and radii rs , rt intersect. Our goal is to design data structures
that maintain the connectivity structure of D(S) as sites are inserted and/or deleted
in S. First, we consider unit disk graphs, i.e., we fix rs = 1, for all sites s ∈ S.
For this case, we describe a data structure that has O(log2 n) amortized update time
and O(log n/ log log n) query time. Second, we look at disk graphs with bounded
radius ratio Ψ, i.e., for all s ∈ S, we have 1 ≤ rs ≤ Ψ, for a parameter Ψ that is
known in advance. Here, we not only investigate the fully dynamic case, but also the
incremental and the decremental scenario, where only insertions or only deletions of
sites are allowed. In the fully dynamic case, we achieve amortized expected update
time O(Ψ log4 n) and query time O(log n/ log log n). This improves the currently
best update time by a factor of Ψ. In the incremental case, we achieve logarithmic
dependency on Ψ, with a data structure that has O(α(n)) amortized query time and
O(log Ψ log4 n) amortized expected update time, where α(n) denotes the inverse Ackermann
function. For the decremental setting, we first develop an efficient decremental
disk revealing data structure: given two sets R and B of disks in the plane, we can delete
disks from B, and upon each deletion, we receive a list of all disks in R that no longer
intersect the union of B. Using this data structure, we get decremental data structures
with a query time of O(log n/ log log n) that supports deletions in O(n log Ψ log4 n)
overall expected time for disk graphs with bounded radius ratio Ψ and O(n log5 n)
overall expected time for disk graphs with arbitrary radii, assuming that the deletion
sequence is oblivious of the internal random choices of the data structures
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Tropical medians by transportation
Fermat-Weber points with respect to an asymmetric tropical distance function
are studied. It turns out that they correspond to the optimal solutions of a
transportation problem. The results are applied to obtain a new method for
computing consensus trees in phylogenetics. This method has several desirable
properties; e.g., it is Pareto and co-Pareto on rooted triplets.Comment: 23 pages, 9 figures, computational experiments adde
Algorithms for Geometric Facility Location: Centers in a Polygon and Dispersion on a Line
We study three geometric facility location problems in this thesis.
First, we consider the dispersion problem in one dimension. We are given an ordered list
of (possibly overlapping) intervals on a line. We wish to choose exactly one point from
each interval such that their left to right ordering on the line matches the input order.
The aim is to choose the points so that the distance between the closest pair of points is
maximized, i.e., they must be socially distanced while respecting the order. We give a new
linear-time algorithm for this problem that produces a lexicographically optimal solution.
We also consider some generalizations of this problem.
For the next two problems, the domain of interest is a simple polygon with n vertices.
The second problem concerns the visibility center. The convention is to think of a polygon
as the top view of a building (or art gallery) where the polygon boundary represents opaque
walls. Two points in the domain are visible to each other if the line segment joining them
does not intersect the polygon exterior. The distance to visibility from a source point to a
target point is the minimum geodesic distance from the source to a point in the polygon
visible to the target. The question is: Where should a single guard be located within the
polygon to minimize the maximum distance to visibility? For m point sites in the polygon,
we give an O((m + n) log (m + n)) time algorithm to determine their visibility center.
Finally, we address the problem of locating the geodesic edge center of a simple polygon—a
point in the polygon that minimizes the maximum geodesic distance to any edge. For a
triangle, this point coincides with its incenter. The geodesic edge center is a generalization
of the well-studied geodesic center (a point that minimizes the maximum distance to any
vertex). Center problems are closely related to farthest Voronoi diagrams, which are well-
studied for point sites in the plane, and less well-studied for line segment sites in the plane.
When the domain is a polygon rather than the whole plane, only the case of point sites has
been addressed—surprisingly, more general sites (with line segments being the simplest
example) have been largely ignored. En route to our solution, we revisit, correct, and
generalize (sometimes in a non-trivial manner) existing algorithms and structures tailored
to work specifically for point sites. We give an optimal linear-time algorithm for finding
the geodesic edge center of a simple polygon
University of Windsor Graduate Calendar 2023 Spring
https://scholar.uwindsor.ca/universitywindsorgraduatecalendars/1027/thumbnail.jp
Drift-diffusion models for innovative semiconductor devices and their numerical solution
We present charge transport models for novel semiconductor devices which may include ionic species as well as their thermodynamically consistent finite volume discretization
Analysing trajectory similarity and improving graph dilation
In this thesis, we focus on two topics in computational geometry. The first topic is analysing trajectory similarity. A trajectory tracks the movement of an object over time. A common way to analyse trajectories is by finding similarities. The Fr\'echet distance is a similarity measure that has gained popularity in the theory community, since it takes the continuity of the curves into account. One way to analyse trajectories using the Fr\'echet distance is to cluster trajectories into groups of similar trajectories. For vehicle trajectories, another way to analyse trajectories is to compute the path on the underlying road network that best represents the trajectory. The second topic is improving graph dilation. Dilation measures the quality of a network in applications such as transportation and communication networks. Spanners are low dilation graphs with not too many edges. Most of the literature on spanners focuses on building the graph from scratch. We instead focus on adding edges to improve the dilation of an existing graph
University of Windsor Graduate Calendar 2023 Winter
https://scholar.uwindsor.ca/universitywindsorgraduatecalendars/1026/thumbnail.jp
Conditional gradients for total variation regularization with PDE constraints: a graph cuts approach
Total variation regularization has proven to be a valuable tool in the
context of optimal control of differential equations. This is particularly
attributed to the observation that TV-penalties often favor piecewise constant
minimizers with well-behaved jumpsets. On the downside, their intricate
properties significantly complicate every aspect of their analysis, from the
derivation of first-order optimality conditions to their discrete approximation
and the choice of a suitable solution algorithm. In this paper, we investigate
a general class of minimization problems with TV-regularization, comprising
both continuous and discretized control spaces, from a convex geometry
perspective. This leads to a variety of novel theoretical insights on
minimization problems with total variation regularization as well as tools for
their practical realization. First, by studying the extremal points of the
respective total variation unit balls, we enable their efficient solution by
geometry exploiting algorithms, e.g. fully-corrective generalized conditional
gradient methods. We give a detailed account on the practical realization of
such a method for piecewise constant finite element approximations of the
control on triangulations of the spatial domain. Second, in the same setting
and for suitable sequences of uniformly refined meshes, it is shown that
minimizers to discretized PDE-constrained optimal control problems approximate
solutions to a continuous limit problem involving an anisotropic total
variation reflecting the fine-scale geometry of the mesh.Comment: 47 pages, 12 figure
Emergent Gravity from the Entanglement Structure in Group Field Theory
We couple a scalar field encoding the entanglement between manifold sites to
group field theory (GFT). The scalar field provides a relational clock that
enables the derivation of the Hamiltonian of the system from the GFT action.
Inspecting this Hamiltonian, we show that a theory of emergent gravity arises,
and that the theory is equivalent to the Ashtekar variables' formulation of
general relativity. The evolution of the system in GFT is a renormalization
group (RG) flow, which corresponds to a simplified Ricci flow, the generator of
which is the Hamiltonian, and the corresponding flow equation is regulated by
the Shroedinger equation. As a consequence of the quantization procedure, the
Hamiltonian is recovered to be non-Hermitian, and can be related to the complex
action formalism, in which the initial conditions and the related future
evolution of the systems are dictated by the imaginary part of the action.Comment: 15 page
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