95,145 research outputs found
A Theory of Stationary Trees and the Balanced Baumgartner-Hajnal-Todorcevic Theorem for Trees
Building on early work by Stevo Todorcevic, we describe a theory of
stationary subtrees of trees of successor-cardinal height. We define the
diagonal union of subsets of a tree, as well as normal ideals on a tree, and we
characterize arbitrary subsets of a non-special tree as being either stationary
or non-stationary.
We then use this theory to prove the following partition relation for trees:
Main Theorem: Let be any infinite regular cardinal, let be any
ordinal such that , and let be any natural
number. Then
This is a generalization to trees of the Balanced
Baumgartner-Hajnal-Todorcevic Theorem, which we recover by applying the above
to the cardinal , the simplest example of a
non--special tree.
As a corollary, we obtain a general result for partially ordered sets:
Theorem: Let be any infinite regular cardinal, let be any
ordinal such that , and let be any natural
number. Let be a partially ordered set such that . Then Comment: Submitted to Acta Mathematica Hungaric
The Balanced Cube: A Concurrent Data Structure
This paper describee the balanced cube, a new data structure for implementing ordered
seta. Conventional dats structures such as heaps, balanced trees and B-trees have root
bottlenecks which limit their potential concurrency and make them unable to take advantage
of the computing potential of concurrent machines. The balanced cube achieves greater
concurrency by eliminating the root bottleneck; an operation in the balanced cube can be
initiated from any node. The throughput of the balanced cube on a concurrent computer is O times O/Log N compared with O(1) for a conventional data structure. Operations on the balanced cube are shown to be deadlock free and consistent with a sequential execution ordered by completion time
Hinted Dictionaries: Efficient Functional Ordered Sets and Maps
This paper introduces hinted dictionaries for expressing efficient ordered sets and maps functionally. As opposed to the traditional ordered dictionaries with logarithmic operations, hinted dictionaries can achieve better performance by using cursor-like objects referred to as hints. Hinted dictionaries unify the interfaces of imperative ordered dictionaries (e.g., C++ maps) and functional ones (e.g., Adams\u27 sets). We show that such dictionaries can use sorted arrays, unbalanced trees, and balanced trees as their underlying representations. Throughout the paper, we use Scala to present the different components of hinted dictionaries. We also provide a C++ implementation to evaluate the effectiveness of hinted dictionaries. Hinted dictionaries provide superior performance for set-set operations in comparison with the standard library of C++. Also, they show a competitive performance in comparison with the SciPy library for sparse vector operations
A new balance index for phylogenetic trees
Several indices that measure the degree of balance of a rooted phylogenetic
tree have been proposed so far in the literature. In this work we define and
study a new index of this kind, which we call the total cophenetic index: the
sum, over all pairs of different leaves, of the depth of their least common
ancestor. This index makes sense for arbitrary trees, can be computed in linear
time and it has a larger range of values and a greater resolution power than
other indices like Colless' or Sackin's. We compute its maximum and minimum
values for arbitrary and binary trees, as well as exact formulas for its
expected value for binary trees under the Yule and the uniform models of
evolution. As a byproduct of this study, we obtain an exact formula for the
expected value of the Sackin index under the uniform model, a result that seems
to be new in the literature.Comment: 24 pages, 2 figures, preliminary version presented at the JBI 201
A determinant formula for the Jones polynomial of pretzel knots
This paper presents an algorithm to construct a weighted adjacency matrix of
a plane bipartite graph obtained from a pretzel knot diagram. The determinant
of this matrix after evaluation is shown to be the Jones polynomial of the
pretzel knot by way of perfect matchings (or dimers) of this graph. The weights
are Tutte's activity letters that arise because the Jones polynomial is a
specialization of the signed version of the Tutte polynomial. The relationship
is formalized between the familiar spanning tree setting for the Tait graph and
the perfect matchings of the plane bipartite graph above. Evaluations of these
activity words are related to the chain complex for the Champanerkar-Kofman
spanning tree model of reduced Khovanov homology.Comment: 19 pages, 12 figures, 2 table
Balanced binary trees in the Tamari lattice
We show that the set of balanced binary trees is closed by interval in the
Tamari lattice. We establish that the intervals [T0, T1] where T0 and T1 are
balanced trees are isomorphic as posets to a hypercube. We introduce tree
patterns and synchronous grammars to get a functional equation of the
generating series enumerating balanced tree intervals
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