177 research outputs found
Pattern Matching for sets of segments
In this paper we present algorithms for a number of problems in geometric
pattern matching where the input consist of a collections of segments in the
plane. Our work consists of two main parts. In the first, we address problems
and measures that relate to collections of orthogonal line segments in the
plane. Such collections arise naturally from problems in mapping buildings and
robot exploration.
We propose a new measure of segment similarity called a \emph{coverage
measure}, and present efficient algorithms for maximising this measure between
sets of axis-parallel segments under translations. Our algorithms run in time
O(n^3\polylog n) in the general case, and run in time O(n^2\polylog n) for
the case when all segments are horizontal. In addition, we show that when
restricted to translations that are only vertical, the Hausdorff distance
between two sets of horizontal segments can be computed in time roughly
O(n^{3/2}{\sl polylog}n). These algorithms form significant improvements over
the general algorithm of Chew et al. that takes time . In the
second part of this paper we address the problem of matching polygonal chains.
We study the well known \Frd, and present the first algorithm for computing the
\Frd under general translations. Our methods also yield algorithms for
computing a generalization of the \Fr distance, and we also present a simple
approximation algorithm for the \Frd that runs in time O(n^2\polylog n).Comment: To appear in the 12 ACM Symposium on Discrete Algorithms, Jan 200
Computing the smallest k-enclosing circle and related problems
AbstractWe present an efficient algorithm for solving the “smallest k-enclosing circle” (kSC) problem: Given a set of n points in the plane and an integer k ⩽ n, find the smallest disk containing k of the points. We present two solutions. When using O(nk) storage, the problem can be solved in time O(nk log2 n). When only O(n log n) storage is allowed, the running time is O(nk log2 n log n/k). We also extend our technique to obtain efficient solutions of several related problems (with similar time and storage bounds). These related problems include: finding the smallest homothetic copy of a given convex polygon P which contains k points from a given planar set, and finding the smallest disk intersecting k segments from a given planar set of non-intersecting segments
Scheduling Sensors for Guaranteed Sparse Coverage
Sensor networks are particularly applicable to the tracking of objects in
motion. For such applications, it may not necessary that the whole region be
covered by sensors as long as the uncovered region is not too large. This
notion has been formalized by Balasubramanian et.al. as the problem of
-weak coverage. This model of coverage provides guarantees about the
regions in which the objects may move undetected. In this paper, we analyse the
theoretical aspects of the problem and provide guarantees about the lifetime
achievable. We introduce a number of practical algorithms and analyse their
significance. The main contribution is a novel linear programming based
algorithm which provides near-optimal lifetime. Through extensive
experimentation, we analyse the performance of these algorithms based on
several parameters defined
Restricted Strip Covering and the Sensor Cover Problem
Given a set of objects with durations (jobs) that cover a base region, can we
schedule the jobs to maximize the duration the original region remains covered?
We call this problem the sensor cover problem. This problem arises in the
context of covering a region with sensors. For example, suppose you wish to
monitor activity along a fence by sensors placed at various fixed locations.
Each sensor has a range and limited battery life. The problem is to schedule
when to turn on the sensors so that the fence is fully monitored for as long as
possible. This one dimensional problem involves intervals on the real line.
Associating a duration to each yields a set of rectangles in space and time,
each specified by a pair of fixed horizontal endpoints and a height. The
objective is to assign a position to each rectangle to maximize the height at
which the spanning interval is fully covered. We call this one dimensional
problem restricted strip covering. If we replace the covering constraint by a
packing constraint, the problem is identical to dynamic storage allocation, a
scheduling problem that is a restricted case of the strip packing problem. We
show that the restricted strip covering problem is NP-hard and present an O(log
log n)-approximation algorithm. We present better approximations or exact
algorithms for some special cases. For the uniform-duration case of restricted
strip covering we give a polynomial-time, exact algorithm but prove that the
uniform-duration case for higher-dimensional regions is NP-hard. Finally, we
consider regions that are arbitrary sets, and we present an O(log
n)-approximation algorithm.Comment: 14 pages, 6 figure
Computing homotopic shortest paths efficiently
AbstractWe give deterministic and randomized algorithms to find shortest paths homotopic to a given collection Π of disjoint paths that wind amongst n point obstacles in the plane. Our deterministic algorithm runs in time O(kout+kinlogn+nn), and the randomized algorithm runs in expected time O(kout+kinlogn+n(logn)1+ε). Here kin is the number of edges in all the paths of Π, and kout is the number of edges in the output paths
Geospatial Tessellation in the Agent-In-Cell Model: A Framework for Agent-Based Modeling of Pandemic
Agent-based simulation is a versatile and potent computational modeling
technique employed to analyze intricate systems and phenomena spanning diverse
fields. However, due to their computational intensity, agent-based models
become more resource-demanding when geographic considerations are introduced.
This study delves into diverse strategies for crafting a series of Agent-Based
Models, named "agent-in-the-cell," which emulate a city. These models,
incorporating geographical attributes of the city and employing real-world
open-source mobility data from Safegraph's publicly available dataset, simulate
the dynamics of COVID spread under varying scenarios. The "agent-in-the-cell"
concept designates that our representative agents, called meta-agents, are
linked to specific home cells in the city's tessellation. We scrutinize
tessellations of the mobility map with varying complexities and experiment with
the agent density, ranging from matching the actual population to reducing the
number of (meta-) agents for computational efficiency. Our findings demonstrate
that tessellations constructed according to the Voronoi Diagram of specific
location types on the street network better preserve dynamics compared to
Census Block Group tessellations and better than Euclidean-based tessellations.
Furthermore, the Voronoi Diagram tessellation and also a hybrid -- Voronoi
Diagram - and Census Block Group - based -- tessellation require fewer
meta-agents to adequately approximate full-scale dynamics. Our analysis spans a
range of city sizes in the United States, encompassing small (Santa Fe, NM),
medium (Seattle, WA), and large (Chicago, IL) urban areas. This examination
also provides valuable insights into the effects of agent count reduction,
varying sensitivity metrics, and the influence of city-specific factors
Geographic max-flow and min-cut under a circular disk failure model
Failures in fiber-optic networks may be caused by natural disasters, such as floods or earthquakes, as well as other events, such as an Electromagnetic Pulse (EMP) attack. These events occur in specific geographical locations, therefore the geography of the network determines the effect of failure events on the network's connectivity and capacity. In this paper we consider a generalization of the min-cut and max-flow problems under a geographic failure model. Specifically, we consider the problem of finding the minimum number of failures, modeled as circular disks, to disconnect a pair of nodes and the maximum number of failure disjoint paths between pairs of nodes. This model applies to the scenario where an adversary is attacking the network multiple times with intention to reduce its connectivity. We present a polynomial time algorithm to solve the geographic min-cut problem and develop an ILP formulation, an exact algorithm, and a heuristic algorithm for the geographic max-flow problem.National Science Foundation (U.S.) (Grant CNS-0830961)National Science Foundation (U.S.) (Grant CNS-1017714)National Science Foundation (U.S.) (Grant CNS-1017800)National Science Foundation (U.S.) (CAREER Grant 0348000)United States. Defense Threat Reduction Agency (Grant HDTRA1-07-1-0004)United States. Defense Threat Reduction Agency (Grant HDTRA-09-1-005
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