14,104 research outputs found
Squarepants in a Tree: Sum of Subtree Clustering and Hyperbolic Pants Decomposition
We provide efficient constant factor approximation algorithms for the
problems of finding a hierarchical clustering of a point set in any metric
space, minimizing the sum of minimimum spanning tree lengths within each
cluster, and in the hyperbolic or Euclidean planes, minimizing the sum of
cluster perimeters. Our algorithms for the hyperbolic and Euclidean planes can
also be used to provide a pants decomposition, that is, a set of disjoint
simple closed curves partitioning the plane minus the input points into subsets
with exactly three boundary components, with approximately minimum total
length. In the Euclidean case, these curves are squares; in the hyperbolic
case, they combine our Euclidean square pants decomposition with our tree
clustering method for general metric spaces.Comment: 22 pages, 14 figures. This version replaces the proof of what is now
Lemma 5.2, as the previous proof was erroneou
Complexity Analysis of Balloon Drawing for Rooted Trees
In a balloon drawing of a tree, all the children under the same parent are
placed on the circumference of the circle centered at their parent, and the
radius of the circle centered at each node along any path from the root
reflects the number of descendants associated with the node. Among various
styles of tree drawings reported in the literature, the balloon drawing enjoys
a desirable feature of displaying tree structures in a rather balanced fashion.
For each internal node in a balloon drawing, the ray from the node to each of
its children divides the wedge accommodating the subtree rooted at the child
into two sub-wedges. Depending on whether the two sub-wedge angles are required
to be identical or not, a balloon drawing can further be divided into two
types: even sub-wedge and uneven sub-wedge types. In the most general case, for
any internal node in the tree there are two dimensions of freedom that affect
the quality of a balloon drawing: (1) altering the order in which the children
of the node appear in the drawing, and (2) for the subtree rooted at each child
of the node, flipping the two sub-wedges of the subtree. In this paper, we give
a comprehensive complexity analysis for optimizing balloon drawings of rooted
trees with respect to angular resolution, aspect ratio and standard deviation
of angles under various drawing cases depending on whether the tree is of even
or uneven sub-wedge type and whether (1) and (2) above are allowed. It turns
out that some are NP-complete while others can be solved in polynomial time. We
also derive approximation algorithms for those that are intractable in general
Colored Non-Crossing Euclidean Steiner Forest
Given a set of -colored points in the plane, we consider the problem of
finding trees such that each tree connects all points of one color class,
no two trees cross, and the total edge length of the trees is minimized. For
, this is the well-known Euclidean Steiner tree problem. For general ,
a -approximation algorithm is known, where is the
Steiner ratio.
We present a PTAS for , a -approximation algorithm
for , and two approximation algorithms for general~, with ratios
and
A near-optimal approximation algorithm for Asymmetric TSP on embedded graphs
We present a near-optimal polynomial-time approximation algorithm for the
asymmetric traveling salesman problem for graphs of bounded orientable or
non-orientable genus. Our algorithm achieves an approximation factor of O(f(g))
on graphs with genus g, where f(n) is the best approximation factor achievable
in polynomial time on arbitrary n-vertex graphs. In particular, the
O(log(n)/loglog(n))-approximation algorithm for general graphs by Asadpour et
al. [SODA 2010] immediately implies an O(log(g)/loglog(g))-approximation
algorithm for genus-g graphs. Our result improves the
O(sqrt(g)*log(g))-approximation algorithm of Oveis Gharan and Saberi [SODA
2011], which applies only to graphs with orientable genus g; ours is the first
approximation algorithm for graphs with bounded non-orientable genus.
Moreover, using recent progress on approximating the genus of a graph, our
O(log(g) / loglog(g))-approximation can be implemented even without an
embedding when the input graph has bounded degree. In contrast, the
O(sqrt(g)*log(g))-approximation algorithm of Oveis Gharan and Saberi requires a
genus-g embedding as part of the input.
Finally, our techniques lead to a O(1)-approximation algorithm for ATSP on
graphs of genus g, with running time 2^O(g)*n^O(1)
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
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