63,139 research outputs found
A distance for partially labeled trees
In a number of practical situations, data have structure and the relations among its component parts need to be coded with suitable data models. Trees are usually utilized for representing data for which hierarchical relations can be defined. This is the case in a number of fields like image analysis, natural language processing, protein structure, or music retrieval, to name a few. In those cases, procedures for comparing trees are very relevant. An approximate tree edit distance algorithm has been introduced for working with trees labeled only at the leaves. In this paper, it has been applied to handwritten character recognition, providing accuracies comparable to those by the most comprehensive search method, being as efficient as the fastest.This work is supported by the Spanish Ministry projects DRIMS (TIN2009-14247-C02), and Consolider Ingenio 2010 (MIPRCV, CSD2007-00018), partially supported by EU ERDF and the Pascal Network of Excellence
A Time Hierarchy Theorem for the LOCAL Model
The celebrated Time Hierarchy Theorem for Turing machines states, informally,
that more problems can be solved given more time. The extent to which a time
hierarchy-type theorem holds in the distributed LOCAL model has been open for
many years. It is consistent with previous results that all natural problems in
the LOCAL model can be classified according to a small constant number of
complexities, such as , etc.
In this paper we establish the first time hierarchy theorem for the LOCAL
model and prove that several gaps exist in the LOCAL time hierarchy.
1. We define an infinite set of simple coloring problems called Hierarchical
-Coloring}. A correctly colored graph can be confirmed by simply
checking the neighborhood of each vertex, so this problem fits into the class
of locally checkable labeling (LCL) problems. However, the complexity of the
-level Hierarchical -Coloring problem is ,
for . The upper and lower bounds hold for both general graphs
and trees, and for both randomized and deterministic algorithms.
2. Consider any LCL problem on bounded degree trees. We prove an
automatic-speedup theorem that states that any randomized -time
algorithm solving the LCL can be transformed into a deterministic -time algorithm. Together with a previous result, this establishes that on
trees, there are no natural deterministic complexities in the ranges
--- or ---.
3. We expose a gap in the randomized time hierarchy on general graphs. Any
randomized algorithm that solves an LCL problem in sublogarithmic time can be
sped up to run in time, which is the complexity of the distributed
Lovasz local lemma problem, currently known to be and
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