2,487 research outputs found
Reachability in Vector Addition Systems is Primitive-Recursive in Fixed Dimension
The reachability problem in vector addition systems is a central question,
not only for the static verification of these systems, but also for many
inter-reducible decision problems occurring in various fields. The currently
best known upper bound on this problem is not primitive-recursive, even when
considering systems of fixed dimension. We provide significant refinements to
the classical decomposition algorithm of Mayr, Kosaraju, and Lambert and to its
termination proof, which yield an ACKERMANN upper bound in the general case,
and primitive-recursive upper bounds in fixed dimension. While this does not
match the currently best known TOWER lower bound for reachability, it is
optimal for related problems
Generic method for bijections between blossoming trees and planar maps
This article presents a unified bijective scheme between planar maps and
blossoming trees, where a blossoming tree is defined as a spanning tree of the
map decorated with some dangling half-edges that enable to reconstruct its
faces. Our method generalizes a previous construction of Bernardi by loosening
its conditions of applications so as to include annular maps, that is maps
embedded in the plane with a root face different from the outer face.
The bijective construction presented here relies deeply on the theory of
\alpha-orientations introduced by Felsner, and in particular on the existence
of minimal and accessible orientations. Since most of the families of maps can
be characterized by such orientations, our generic bijective method is proved
to capture as special cases all previously known bijections involving
blossoming trees: for example Eulerian maps, m-Eulerian maps, non separable
maps and simple triangulations and quadrangulations of a k-gon. Moreover, it
also permits to obtain new bijective constructions for bipolar orientations and
d-angulations of girth d of a k-gon.
As for applications, each specialization of the construction translates into
enumerative by-products, either via a closed formula or via a recursive
computational scheme. Besides, for every family of maps described in the paper,
the construction can be implemented in linear time. It yields thus an effective
way to encode and generate planar maps.
In a recent work, Bernardi and Fusy introduced another unified bijective
scheme, we adopt here a different strategy which allows us to capture different
bijections. These two approaches should be seen as two complementary ways of
unifying bijections between planar maps and decorated trees.Comment: 45 pages, comments welcom
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