39,246 research outputs found
Exotic trees
We discuss the scaling properties of free branched polymers. The scaling
behaviour of the model is classified by the Hausdorff dimensions for the
internal geometry: d_L and d_H, and for the external one: D_L and D_H. The
dimensions d_H and D_H characterize the behaviour for long distances while d_L
and D_L for short distances. We show that the internal Hausdorff dimension is
d_L=2 for generic and scale-free trees, contrary to d_H which is known be equal
two for generic trees and to vary between two and infinity for scale-free
trees. We show that the external Hausdorff dimension D_H is directly related to
the internal one as D_H = \alpha d_H, where \alpha is the stability index of
the embedding weights for the nearest-vertex interactions. The index is
\alpha=2 for weights from the gaussian domain of attraction and 0<\alpha <2 for
those from the L\'evy domain of attraction. If the dimension D of the target
space is larger than D_H one finds D_L=D_H, or otherwise D_L=D. The latter
result means that the fractal structure cannot develop in a target space which
has too low dimension.Comment: 33 pages, 6 eps figure
Decomposition of Trees and Paths via Correlation
We study the problem of decomposing (clustering) a tree with respect to costs
attributed to pairs of nodes, so as to minimize the sum of costs for those
pairs of nodes that are in the same component (cluster). For the general case
and for the special case of the tree being a star, we show that the problem is
NP-hard. For the special case of the tree being a path, this problem is known
to be polynomial time solvable. We characterize several classes of facets of
the combinatorial polytope associated with a formulation of this clustering
problem in terms of lifted multicuts. In particular, our results yield a
complete totally dual integral (TDI) description of the lifted multicut
polytope for paths, which establishes a connection to the combinatorial
properties of alternative formulations such as set partitioning.Comment: v2 is a complete revisio
Lossless and near-lossless source coding for multiple access networks
A multiple access source code (MASC) is a source code designed for the following network configuration: a pair of correlated information sequences {X-i}(i=1)(infinity), and {Y-i}(i=1)(infinity) is drawn independent and identically distributed (i.i.d.) according to joint probability mass function (p.m.f.) p(x, y); the encoder for each source operates without knowledge of the other source; the decoder jointly decodes the encoded bit streams from both sources. The work of Slepian and Wolf describes all rates achievable by MASCs of infinite coding dimension (n --> infinity) and asymptotically negligible error probabilities (P-e((n)) --> 0). In this paper, we consider the properties of optimal instantaneous MASCs with finite coding dimension (n 0) performance. The interest in near-lossless codes is inspired by the discontinuity in the limiting rate region at P-e((n)) = 0 and the resulting performance benefits achievable by using near-lossless MASCs as entropy codes within lossy MASCs. Our central results include generalizations of Huffman and arithmetic codes to the MASC framework for arbitrary p(x, y), n, and P-e((n)) and polynomial-time design algorithms that approximate these optimal solutions
Recognizing Partial Cubes in Quadratic Time
We show how to test whether a graph with n vertices and m edges is a partial
cube, and if so how to find a distance-preserving embedding of the graph into a
hypercube, in the near-optimal time bound O(n^2), improving previous O(nm)-time
solutions.Comment: 25 pages, five figures. This version significantly expands previous
versions, including a new report on an implementation of the algorithm and
experiments with i
Setting Parameters by Example
We introduce a class of "inverse parametric optimization" problems, in which
one is given both a parametric optimization problem and a desired optimal
solution; the task is to determine parameter values that lead to the given
solution. We describe algorithms for solving such problems for minimum spanning
trees, shortest paths, and other "optimal subgraph" problems, and discuss
applications in multicast routing, vehicle path planning, resource allocation,
and board game programming.Comment: 13 pages, 3 figures. To be presented at 40th IEEE Symp. Foundations
of Computer Science (FOCS '99
Geometric combinatorial algebras: cyclohedron and simplex
In this paper we report on results of our investigation into the algebraic
structure supported by the combinatorial geometry of the cyclohedron. Our new
graded algebra structures lie between two well known Hopf algebras: the
Malvenuto-Reutenauer algebra of permutations and the Loday-Ronco algebra of
binary trees. Connecting algebra maps arise from a new generalization of the
Tonks projection from the permutohedron to the associahedron, which we discover
via the viewpoint of the graph associahedra of Carr and Devadoss. At the same
time that viewpoint allows exciting geometrical insights into the
multiplicative structure of the algebras involved. Extending the Tonks
projection also reveals a new graded algebra structure on the simplices.
Finally this latter is extended to a new graded Hopf algebra (one-sided) with
basis all the faces of the simplices.Comment: 23 figures, new expanded section about Hopf algebra of simplices,
with journal correction
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