17,417 research outputs found
On factorisation forests
The theorem of factorisation forests shows the existence of nested
factorisations -- a la Ramsey -- for finite words. This theorem has important
applications in semigroup theory, and beyond. The purpose of this paper is to
illustrate the importance of this approach in the context of automata over
infinite words and trees. We extend the theorem of factorisation forest in two
directions: we show that it is still valid for any word indexed by a linear
ordering; and we show that it admits a deterministic variant for words indexed
by well-orderings. A byproduct of this work is also an improvement on the known
bounds for the original result. We apply the first variant for giving a
simplified proof of the closure under complementation of rational sets of words
indexed by countable scattered linear orderings. We apply the second variant in
the analysis of monadic second-order logic over trees, yielding new results on
monadic interpretations over trees. Consequences of it are new caracterisations
of prefix-recognizable structures and of the Caucal hierarchy.Comment: 27 page
Embedding large subgraphs into dense graphs
What conditions ensure that a graph G contains some given spanning subgraph
H? The most famous examples of results of this kind are probably Dirac's
theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect
matchings are generalized by perfect F-packings, where instead of covering all
the vertices of G by disjoint edges, we want to cover G by disjoint copies of a
(small) graph F. It is unlikely that there is a characterization of all graphs
G which contain a perfect F-packing, so as in the case of Dirac's theorem it
makes sense to study conditions on the minimum degree of G which guarantee a
perfect F-packing.
The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy
and Szemeredi have proved to be powerful tools in attacking such problems and
quite recently, several long-standing problems and conjectures in the area have
been solved using these. In this survey, we give an outline of recent progress
(with our main emphasis on F-packings, Hamiltonicity problems and tree
embeddings) and describe some of the methods involved
Efficient Construction of Probabilistic Tree Embeddings
In this paper we describe an algorithm that embeds a graph metric
on an undirected weighted graph into a distribution of tree metrics
such that for every pair , and
. Such embeddings have
proved highly useful in designing fast approximation algorithms, as many hard
problems on graphs are easy to solve on tree instances. For a graph with
vertices and edges, our algorithm runs in time with high
probability, which improves the previous upper bound of shown by
Mendel et al.\,in 2009.
The key component of our algorithm is a new approximate single-source
shortest-path algorithm, which implements the priority queue with a new data
structure, the "bucket-tree structure". The algorithm has three properties: it
only requires linear time in the number of edges in the input graph; the
computed distances have a distance preserving property; and when computing the
shortest-paths to the -nearest vertices from the source, it only requires to
visit these vertices and their edge lists. These properties are essential to
guarantee the correctness and the stated time bound.
Using this shortest-path algorithm, we show how to generate an intermediate
structure, the approximate dominance sequences of the input graph, in time, and further propose a simple yet efficient algorithm to converted
this sequence to a tree embedding in time, both with high
probability. Combining the three subroutines gives the stated time bound of the
algorithm.
Then we show that this efficient construction can facilitate some
applications. We proved that FRT trees (the generated tree embedding) are
Ramsey partitions with asymptotically tight bound, so the construction of a
series of distance oracles can be accelerated
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