183 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
Proximity Drawings of High-Degree Trees
A drawing of a given (abstract) tree that is a minimum spanning tree of the
vertex set is considered aesthetically pleasing. However, such a drawing can
only exist if the tree has maximum degree at most 6. What can be said for trees
of higher degree? We approach this question by supposing that a partition or
covering of the tree by subtrees of bounded degree is given. Then we show that
if the partition or covering satisfies some natural properties, then there is a
drawing of the entire tree such that each of the given subtrees is drawn as a
minimum spanning tree of its vertex set
Learning loopy graphical models with latent variables: Efficient methods and guarantees
The problem of structure estimation in graphical models with latent variables
is considered. We characterize conditions for tractable graph estimation and
develop efficient methods with provable guarantees. We consider models where
the underlying Markov graph is locally tree-like, and the model is in the
regime of correlation decay. For the special case of the Ising model, the
number of samples required for structural consistency of our method scales
as , where p is the
number of variables, is the minimum edge potential, is
the depth (i.e., distance from a hidden node to the nearest observed nodes),
and is a parameter which depends on the bounds on node and edge
potentials in the Ising model. Necessary conditions for structural consistency
under any algorithm are derived and our method nearly matches the lower bound
on sample requirements. Further, the proposed method is practical to implement
and provides flexibility to control the number of latent variables and the
cycle lengths in the output graph.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1070 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Parallel Algorithms for Geometric Graph Problems
We give algorithms for geometric graph problems in the modern parallel models
inspired by MapReduce. For example, for the Minimum Spanning Tree (MST) problem
over a set of points in the two-dimensional space, our algorithm computes a
-approximate MST. Our algorithms work in a constant number of
rounds of communication, while using total space and communication proportional
to the size of the data (linear space and near linear time algorithms). In
contrast, for general graphs, achieving the same result for MST (or even
connectivity) remains a challenging open problem, despite drawing significant
attention in recent years.
We develop a general algorithmic framework that, besides MST, also applies to
Earth-Mover Distance (EMD) and the transportation cost problem. Our algorithmic
framework has implications beyond the MapReduce model. For example it yields a
new algorithm for computing EMD cost in the plane in near-linear time,
. We note that while recently Sharathkumar and Agarwal
developed a near-linear time algorithm for -approximating EMD,
our algorithm is fundamentally different, and, for example, also solves the
transportation (cost) problem, raised as an open question in their work.
Furthermore, our algorithm immediately gives a -approximation
algorithm with space in the streaming-with-sorting model with
passes. As such, it is tempting to conjecture that the
parallel models may also constitute a concrete playground in the quest for
efficient algorithms for EMD (and other similar problems) in the vanilla
streaming model, a well-known open problem
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