24,143 research outputs found
Dispersing Points on Intervals
We consider a problem of dispersing points on disjoint intervals on a line. Given n pairwise disjoint intervals sorted on a line, we want to find a point in each interval such that the minimum pairwise distance of these points is maximized. Based on a greedy strategy, we present a linear time algorithm for the problem. Further, we also solve in linear time the cycle version of the problem where the intervals are given on a cycle
Geometric Algorithms for Intervals and Related Problems
In this dissertation, we study several problems related to intervals and develop efficient algorithms for them. Interval problems have many applications in reality because many objects, values, and ranges are intervals in nature, such as time intervals, distances, line segments, probabilities, etc. Problems on intervals are gaining attention also because intervals are among the most basic geometric objects, and for the same reason, computational geometry techniques find useful for attacking these problems. Specifically, the problems we study in this dissertation includes the following: balanced splitting on weighted intervals, minimizing the movements of spreading points, dispersing points on intervals, multiple barrier coverage, and separating overlapped intervals on a line. We develop efficient algorithms for these problems and our results are either first known solutions or improve the previous work.
In the problem of balanced splitting on weighted intervals, we are given a set of n intervals with non-negative weights on a line and an integer k ≥ 1. The goal is to find k points to partition the line into k + 1 segments, such that the maximum sum of the interval weights in these segments is minimized. We give an algorithm that solves the problem in O(n log n) time. Our second problem is on minimizing the movements of spreading points. In this problem, we are given a set of points on a line and we want to spread the points on the line so that the minimum pairwise distance of all points is no smaller than a given value δ. The objective is to minimize the maximum moving distance of all points. We solve the problem in O(n) time. We also solve the cycle version of the problem in linear time. For the third problem, we are given a set of n non-overlapping intervals on a line and we want to place a point on each interval so that the minimum pairwise distance of all points are maximized. We present an O(n) time algorithm for the problem. We also solve its cycle version in O(n) time. The fourth problem is on multiple barrier coverage, where we are given n sensors in the plane and m barriers (represented by intervals) on a line. The goal is to move the sensors onto the line to cover all the barriers such that the maximum moving distance of all sensors is minimized. Our algorithm for the problem runs in O(n2 log n log log n + nm log m) time. In a special case where the sensors are all initially on the line, our algorithm runs in O((n + m) log(n + m)) time. Finally, for the problem of separating overlapped intervals, we have a set of n intervals (possibly overlapped) on a line and we want to move them along the line so that no two intervals properly intersect. The objective is to minimize the maximum moving distance of all intervals. We propose an O(n log n) time algorithm for the problem.
The algorithms and techniques developed in this dissertation are quite basic and fundamental, so they might be useful for solving other related problems on intervals as well
The effects of landscape modifications on the long-term persistence of animal populations
Background: The effects of landscape modifications on the long-term persistence of wild animal populations is of crucial
importance to wildlife managers and conservation biologists, but obtaining experimental evidence using real landscapes is
usually impossible. To circumvent this problem we used individual-based models (IBMs) of interacting animals in
experimental modifications of a real Danish landscape. The models incorporate as much as possible of the behaviour and
ecology of four species with contrasting life-history characteristics: skylark (Alauda arvensis), vole (Microtus agrestis), a
ground beetle (Bembidion lampros) and a linyphiid spider (Erigone atra). This allows us to quantify the population
implications of experimental modifications of landscape configuration and composition.
Methodology/Principal Findings: Starting with a real agricultural landscape, we progressively reduced landscape
complexity by (i) homogenizing habitat patch shapes, (ii) randomizing the locations of the patches, and (iii) randomizing the
size of the patches. The first two steps increased landscape fragmentation. We assessed the effects of these manipulations
on the long-term persistence of animal populations by measuring equilibrium population sizes and time to recovery after
disturbance. Patch rearrangement and the presence of corridors had a large effect on the population dynamics of species
whose local success depends on the surrounding terrain. Landscape modifications that reduced population sizes increased
recovery times in the short-dispersing species, making small populations vulnerable to increasing disturbance. The species
that were most strongly affected by large disturbances fluctuated little in population sizes in years when no perturbations
took place.
Significance: Traditional approaches to the management and conservation of populations use either classical methods of
population analysis, which fail to adequately account for the spatial configurations of landscapes, or landscape ecology,
which accounts for landscape structure but has difficulty predicting the dynamics of populations living in them. Here we
show how realistic and replicable individual-based models can bridge the gap between non-spatial population theory and
non-dynamic landscape ecology. A major strength of the approach is its ability to identify population vulnerabilities not
detected by standard population viability analyses
Billiards with polynomial mixing rates
While many dynamical systems of mechanical origin, in particular billiards,
are strongly chaotic -- enjoy exponential mixing, the rates of mixing in many
other models are slow (algebraic, or polynomial). The dynamics in the latter
are intermittent between regular and chaotic, which makes them particularly
interesting in physical studies. However, mathematical methods for the analysis
of systems with slow mixing rates were developed just recently and are still
difficult to apply to realistic models. Here we reduce those methods to a
practical scheme that allows us to obtain a nearly optimal bound on mixing
rates. We demonstrate how the method works by applying it to several classes of
chaotic billiards with slow mixing as well as discuss a few examples where the
method, in its present form, fails.Comment: 39pages, 11 figue
Symbolic dynamics I. Finite dispersive billiards
Orbits in different dispersive billiard systems, e.g. the 3 disk system, are
mapped into a topological well ordered symbol plane and it is showed that
forbidden and allowed orbits are separated by a monotone pruning front. The
pruning front can be approximated by a sequence of finite symbolic dynamics
grammars.Comment: CYCLER Paper 93Jan00
Non-ergodicity of the motion in three dimensional steep repelling dispersing potentials
It is demonstrated numerically that smooth three degrees of freedom
Hamiltonian systems which are arbitrarily close to three dimensional strictly
dispersing billiards (Sinai billiards) have islands of effective stability, and
hence are non-ergodic. The mechanism for creating the islands are corners of
the billiard domain.Comment: 6 pages, 8 figures, submitted to Chao
Recommended from our members
Concentration fluctuations in atmospheric dispersion
This report summarizes work done at Brunel University under Agreement No.2066/62 from
15 July 1986 to 14 July 1989. The title of the project was Concentration Fluctuations in Atmospheric Dispersion. The report has three principal components. These are:
(i) theoretical work on the electrostatic effects associated with dispersing charged tracers.
(ii) extensive analysis of several datasets taken with the CDE sensor system, particularly one obtained at RAF Cardington on 10 May 1988;
(iii) interpretation of the results of the analysis.
The conclusions of the report include recommendations for further work to exploit the
advantages that the system has over many others
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