10,315 research outputs found
Effective Scalar Products for D-finite Symmetric Functions
Many combinatorial generating functions can be expressed as combinations of
symmetric functions, or extracted as sub-series and specializations from such
combinations. Gessel has outlined a large class of symmetric functions for
which the resulting generating functions are D-finite. We extend Gessel's work
by providing algorithms that compute differential equations these generating
functions satisfy in the case they are given as a scalar product of symmetric
functions in Gessel's class. Examples of applications to k-regular graphs and
Young tableaux with repeated entries are given. Asymptotic estimates are a
natural application of our method, which we illustrate on the same model of
Young tableaux. We also derive a seemingly new formula for the Kronecker
product of the sum of Schur functions with itself.Comment: 51 pages, full paper version of FPSAC 02 extended abstract; v2:
corrections from original submission, improved clarity; now formatted for
journal + bibliograph
Random sampling of plane partitions
This article presents uniform random generators of plane partitions according
to the size (the number of cubes in the 3D interpretation). Combining a
bijection of Pak with the method of Boltzmann sampling, we obtain random
samplers that are slightly superlinear: the complexity is in
approximate-size sampling and in exact-size sampling
(under a real-arithmetic computation model). To our knowledge, these are the
first polynomial-time samplers for plane partitions according to the size
(there exist polynomial-time samplers of another type, which draw plane
partitions that fit inside a fixed bounding box). The same principles yield
efficient samplers for -boxed plane partitions (plane partitions
with two dimensions bounded), and for skew plane partitions. The random
samplers allow us to perform simulations and observe limit shapes and frozen
boundaries, which have been analysed recently by Cerf and Kenyon for plane
partitions, and by Okounkov and Reshetikhin for skew plane partitions.Comment: 23 page
Perfect sampling algorithm for Schur processes
We describe random generation algorithms for a large class of random
combinatorial objects called Schur processes, which are sequences of random
(integer) partitions subject to certain interlacing conditions. This class
contains several fundamental combinatorial objects as special cases, such as
plane partitions, tilings of Aztec diamonds, pyramid partitions and more
generally steep domino tilings of the plane. Our algorithm, which is of
polynomial complexity, is both exact (i.e. the output follows exactly the
target probability law, which is either Boltzmann or uniform in our case), and
entropy optimal (i.e. it reads a minimal number of random bits as an input).
The algorithm encompasses previous growth procedures for special Schur
processes related to the primal and dual RSK algorithm, as well as the famous
domino shuffling algorithm for domino tilings of the Aztec diamond. It can be
easily adapted to deal with symmetric Schur processes and general Schur
processes involving infinitely many parameters. It is more concrete and easier
to implement than Borodin's algorithm, and it is entropy optimal.
At a technical level, it relies on unified bijective proofs of the different
types of Cauchy and Littlewood identities for Schur functions, and on an
adaptation of Fomin's growth diagram description of the RSK algorithm to that
setting. Simulations performed with this algorithm suggest interesting limit
shape phenomena for the corresponding tiling models, some of which are new.Comment: 26 pages, 19 figures (v3: final version, corrected a few misprints
present in v2
Fixed-Parameter Algorithms for Rectilinear Steiner tree and Rectilinear Traveling Salesman Problem in the plane
Given a set of points with their pairwise distances, the traveling
salesman problem (TSP) asks for a shortest tour that visits each point exactly
once. A TSP instance is rectilinear when the points lie in the plane and the
distance considered between two points is the distance. In this paper, a
fixed-parameter algorithm for the Rectilinear TSP is presented and relies on
techniques for solving TSP on bounded-treewidth graphs. It proves that the
problem can be solved in where denotes the
number of horizontal lines containing the points of . The same technique can
be directly applied to the problem of finding a shortest rectilinear Steiner
tree that interconnects the points of providing a
time complexity. Both bounds improve over the best time bounds known for these
problems.Comment: 24 pages, 13 figures, 6 table
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