11,192 research outputs found
A Framework for Algorithm Stability
We say that an algorithm is stable if small changes in the input result in
small changes in the output. This kind of algorithm stability is particularly
relevant when analyzing and visualizing time-varying data. Stability in general
plays an important role in a wide variety of areas, such as numerical analysis,
machine learning, and topology, but is poorly understood in the context of
(combinatorial) algorithms. In this paper we present a framework for analyzing
the stability of algorithms. We focus in particular on the tradeoff between the
stability of an algorithm and the quality of the solution it computes. Our
framework allows for three types of stability analysis with increasing degrees
of complexity: event stability, topological stability, and Lipschitz stability.
We demonstrate the use of our stability framework by applying it to kinetic
Euclidean minimum spanning trees
Predicting Whole Forest Structure, Primary Productivity, and Biomass Density From Maximum Tree Size and Resource Limitations
In the face of uncertain biological response to climate change and the many
critiques concerning model complexity it is increasingly important to develop
predictive mechanistic frameworks that capture the dominant features of
ecological communities and their dependencies on environmental factors. This is
particularly important for critical global processes such as biomass changes,
carbon export, and biogenic climate feedback. Past efforts have successfully
understood a broad spectrum of plant and community traits across a range of
biological diversity and body size, including tree size distributions and
maximum tree height, from mechanical, hydrodynamic, and resource constraints.
Recently it was shown that global scaling relationships for net primary
productivity are correlated with local meteorology and the overall biomass
density within a forest. Along with previous efforts, this highlights the
connection between widely observed allometric relationships and predictive
ecology. An emerging goal of ecological theory is to gain maximum predictive
power with the least number of parameters. Here we show that the explicit
dependence of such critical quantities can be systematically predicted knowing
just the size of the largest tree. This is supported by data showing that
forests converge to our predictions as they mature. Since maximum tree size can
be calculated from local meteorology this provides a general framework for
predicting the generic structure of forests from local environmental parameters
thereby addressing a range of critical Earth-system questions.Comment: 26 pages, 4 figures, 1 Tabl
Agglomerative Clustering of Growing Squares
We study an agglomerative clustering problem motivated by interactive glyphs
in geo-visualization. Consider a set of disjoint square glyphs on an
interactive map. When the user zooms out, the glyphs grow in size relative to
the map, possibly with different speeds. When two glyphs intersect, we wish to
replace them by a new glyph that captures the information of the intersecting
glyphs.
We present a fully dynamic kinetic data structure that maintains a set of
disjoint growing squares. Our data structure uses
space, supports queries in worst case time, and updates in
amortized time. This leads to an time
algorithm to solve the agglomerative clustering problem. This is a significant
improvement over the current best time algorithms.Comment: 14 pages, 7 figure
Globally and Locally Minimal Weight Spanning Tree Networks
The competition between local and global driving forces is significant in a
wide variety of naturally occurring branched networks. We have investigated the
impact of a global minimization criterion versus a local one on the structure
of spanning trees. To do so, we consider two spanning tree structures - the
generalized minimal spanning tree (GMST) defined by Dror et al. [1] and an
analogous structure based on the invasion percolation network, which we term
the generalized invasive spanning tree or GIST. In general, these two
structures represent extremes of global and local optimality, respectively.
Structural characteristics are compared between the GMST and GIST for a fixed
lattice. In addition, we demonstrate a method for creating a series of
structures which enable one to span the range between these two extremes. Two
structural characterizations, the occupied edge density (i.e., the fraction of
edges in the graph that are included in the tree) and the tortuosity of the
arcs in the trees, are shown to correlate well with the degree to which an
intermediate structure resembles the GMST or GIST. Both characterizations are
straightforward to determine from an image and are potentially useful tools in
the analysis of the formation of network structures.Comment: 23 pages, 5 figures, 2 tables, typographical error correcte
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