56,361 research outputs found
The Number of Triangles Needed to Span a Polygon Embedded in R^d
Given a closed polygon P having n edges, embedded in R^d, we give upper and
lower bounds for the minimal number of triangles t needed to form a
triangulated PL surface in R^d having P as its geometric boundary. The most
interesting case is dimension 3, where the polygon may be knotted. We use the
Seifert suface construction to show there always exists an embedded surface
requiring at most 7n^2 triangles. We complement this result by showing there
are polygons in R^3 for which any embedded surface requires at least 1/2n^2 -
O(n) triangles. In dimension 2 only n-2 triangles are needed, and in dimensions
5 or more there exists an embedded surface requiring at most n triangles. In
dimension 4 we obtain a partial answer, with an O(n^2) upper bound for embedded
surfaces, and a construction of an immersed disk requiring at most 3n
triangles. These results can be interpreted as giving qualitiative discrete
analogues of the isoperimetric inequality for piecewise linear manifolds.Comment: 16 pages, 4 figures. This paper is a retitled, revised version of
math.GT/020217
Nowhere Dense Graph Classes and Dimension
Nowhere dense graph classes provide one of the least restrictive notions of
sparsity for graphs. Several equivalent characterizations of nowhere dense
classes have been obtained over the years, using a wide range of combinatorial
objects. In this paper we establish a new characterization of nowhere dense
classes, in terms of poset dimension: A monotone graph class is nowhere dense
if and only if for every and every , posets of height
at most with elements and whose cover graphs are in the class have
dimension .Comment: v4: Minor changes suggested by a refere
SqFreeEVAL: An (almost) optimal real-root isolation algorithm
Let f be a univariate polynomial with real coefficients, f in R[X].
Subdivision algorithms based on algebraic techniques (e.g., Sturm or Descartes
methods) are widely used for isolating the real roots of f in a given interval.
In this paper, we consider a simple subdivision algorithm whose primitives are
purely numerical (e.g., function evaluation). The complexity of this algorithm
is adaptive because the algorithm makes decisions based on local data. The
complexity analysis of adaptive algorithms (and this algorithm in particular)
is a new challenge for computer science. In this paper, we compute the size of
the subdivision tree for the SqFreeEVAL algorithm.
The SqFreeEVAL algorithm is an evaluation-based numerical algorithm which is
well-known in several communities. The algorithm itself is simple, but prior
attempts to compute its complexity have proven to be quite technical and have
yielded sub-optimal results. Our main result is a simple O(d(L+ln d)) bound on
the size of the subdivision tree for the SqFreeEVAL algorithm on the benchmark
problem of isolating all real roots of an integer polynomial f of degree d and
whose coefficients can be written with at most L bits.
Our proof uses two amortization-based techniques: First, we use the algebraic
amortization technique of the standard Mahler-Davenport root bounds to
interpret the integral in terms of d and L. Second, we use a continuous
amortization technique based on an integral to bound the size of the
subdivision tree. This paper is the first to use the novel analysis technique
of continuous amortization to derive state of the art complexity bounds
Bounds on the Coefficients of Tension and Flow Polynomials
The goal of this article is to obtain bounds on the coefficients of modular
and integral flow and tension polynomials of graphs. To this end we make use of
the fact that these polynomials can be realized as Ehrhart polynomials of
inside-out polytopes. Inside-out polytopes come with an associated relative
polytopal complex and, for a wide class of inside-out polytopes, we show that
this complex has a convex ear decomposition. This leads to the desired bounds
on the coefficients of these polynomials.Comment: 16 page
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