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In this article we present a method to generate random objects from a large variety of combinatorial classes according to a given distribution. Given a description of the combinatorial class and a set of sample data our method will provide an algorithm that generates objects of size n in worst-case runtime O(n^2) (O(n log(n)) can be achieved at the cost of a higher average-case runtime), with the generated objects following a distribution that closely matches the distribution of the sample data.

For many decades, the search for language classes that extend the
context-free laguages enough to include various languages that arise in
practice, while still keeping as many of the useful properties that
context-free grammars have - most notably cubic parsing time - has been
one of the major areas of research in formal language theory. In this thesis
we add a new family of classes to this field, namely
position-and-length-dependent context-free grammars. Our classes use the
approach of regulated rewriting, where derivations in a context-free base
grammar are allowed or forbidden based on, e.g., the sequence of rules used
in a derivation or the sentential forms, each rule is applied to. For our
new classes we look at the yield of each rule application, i.e. the
subword of the final word that eventually is derived from the symbols
introduced by the rule application. The position and length of the yield
in the final word define the position and length of the rule application and
each rule is associated a set of positions and lengths where it is allowed
to be applied.
We show that - unless the sets of allowed positions and lengths are really
complex - the languages in our classes can be parsed in the same time as
context-free grammars, using slight adaptations of well-known parsing
algorithms. We also show that they form a proper hierarchy above the
context-free languages and examine their relation to language classes
defined by other types of regulated rewriting.
We complete the treatment of the language classes by introducing pushdown
automata with position counter, an extension of traditional pushdown
automata that recognizes the languages generated by
position-and-length-dependent context-free grammars, and we examine various
closure and decidability properties of our classes. Additionally, we gather
the corresponding results for the subclasses that use right-linear resp.
left-linear base grammars and the corresponding class of automata, finite
automata with position counter.
Finally, as an application of our idea, we introduce length-dependent
stochastic context-free grammars and show how they can be employed to
improve the quality of predictions for RNA secondary structures.