2,574 research outputs found
Fast Hierarchical Clustering and Other Applications of Dynamic Closest Pairs
We develop data structures for dynamic closest pair problems with arbitrary
distance functions, that do not necessarily come from any geometric structure
on the objects. Based on a technique previously used by the author for
Euclidean closest pairs, we show how to insert and delete objects from an
n-object set, maintaining the closest pair, in O(n log^2 n) time per update and
O(n) space. With quadratic space, we can instead use a quadtree-like structure
to achieve an optimal time bound, O(n) per update. We apply these data
structures to hierarchical clustering, greedy matching, and TSP heuristics, and
discuss other potential applications in machine learning, Groebner bases, and
local improvement algorithms for partition and placement problems. Experiments
show our new methods to be faster in practice than previously used heuristics.Comment: 20 pages, 9 figures. A preliminary version of this paper appeared at
the 9th ACM-SIAM Symp. on Discrete Algorithms, San Francisco, 1998, pp.
619-628. For source code and experimental results, see
http://www.ics.uci.edu/~eppstein/projects/pairs
The cavity approach for Steiner trees packing problems
The Belief Propagation approximation, or cavity method, has been recently
applied to several combinatorial optimization problems in its zero-temperature
implementation, the max-sum algorithm. In particular, recent developments to
solve the edge-disjoint paths problem and the prize-collecting Steiner tree
problem on graphs have shown remarkable results for several classes of graphs
and for benchmark instances. Here we propose a generalization of these
techniques for two variants of the Steiner trees packing problem where multiple
"interacting" trees have to be sought within a given graph. Depending on the
interaction among trees we distinguish the vertex-disjoint Steiner trees
problem, where trees cannot share nodes, from the edge-disjoint Steiner trees
problem, where edges cannot be shared by trees but nodes can be members of
multiple trees. Several practical problems of huge interest in network design
can be mapped into these two variants, for instance, the physical design of
Very Large Scale Integration (VLSI) chips. The formalism described here relies
on two components edge-variables that allows us to formulate a massage-passing
algorithm for the V-DStP and two algorithms for the E-DStP differing in the
scaling of the computational time with respect to some relevant parameters. We
will show that one of the two formalisms used for the edge-disjoint variant
allow us to map the max-sum update equations into a weighted maximum matching
problem over proper bipartite graphs. We developed a heuristic procedure based
on the max-sum equations that shows excellent performance in synthetic networks
(in particular outperforming standard multi-step greedy procedures by large
margins) and on large benchmark instances of VLSI for which the optimal
solution is known, on which the algorithm found the optimum in two cases and
the gap to optimality was never larger than 4 %
Sparse Representation of Astronomical Images
Sparse representation of astronomical images is discussed. It is shown that a
significant gain in sparsity is achieved when particular mixed dictionaries are
used for approximating these types of images with greedy selection strategies.
Experiments are conducted to confirm: i)Effectiveness at producing sparse
representations. ii)Competitiveness, with respect to the time required to
process large images.The latter is a consequence of the suitability of the
proposed dictionaries for approximating images in partitions of small
blocks.This feature makes it possible to apply the effective greedy selection
technique Orthogonal Matching Pursuit, up to some block size. For blocks
exceeding that size a refinement of the original Matching Pursuit approach is
considered. The resulting method is termed Self Projected Matching Pursuit,
because is shown to be effective for implementing, via Matching Pursuit itself,
the optional back-projection intermediate steps in that approach.Comment: Software to implement the approach is available on
http://www.nonlinear-approx.info/examples/node1.htm
Single-Strip Triangulation of Manifolds with Arbitrary Topology
Triangle strips have been widely used for efficient rendering. It is
NP-complete to test whether a given triangulated model can be represented as a
single triangle strip, so many heuristics have been proposed to partition
models into few long strips. In this paper, we present a new algorithm for
creating a single triangle loop or strip from a triangulated model. Our method
applies a dual graph matching algorithm to partition the mesh into cycles, and
then merges pairs of cycles by splitting adjacent triangles when necessary. New
vertices are introduced at midpoints of edges and the new triangles thus formed
are coplanar with their parent triangles, hence the visual fidelity of the
geometry is not changed. We prove that the increase in the number of triangles
due to this splitting is 50% in the worst case, however for all models we
tested the increase was less than 2%. We also prove tight bounds on the number
of triangles needed for a single-strip representation of a model with holes on
its boundary. Our strips can be used not only for efficient rendering, but also
for other applications including the generation of space filling curves on a
manifold of any arbitrary topology.Comment: 12 pages, 10 figures. To appear at Eurographics 200
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