2,013 research outputs found
Metric Embedding via Shortest Path Decompositions
We study the problem of embedding shortest-path metrics of weighted graphs
into spaces. We introduce a new embedding technique based on low-depth
decompositions of a graph via shortest paths. The notion of Shortest Path
Decomposition depth is inductively defined: A (weighed) path graph has shortest
path decomposition (SPD) depth . General graph has an SPD of depth if it
contains a shortest path whose deletion leads to a graph, each of whose
components has SPD depth at most . In this paper we give an
-distortion embedding for graphs of SPD
depth at most . This result is asymptotically tight for any fixed ,
while for it is tight up to second order terms.
As a corollary of this result, we show that graphs having pathwidth embed
into with distortion . For
, this improves over the best previous bound of Lee and Sidiropoulos that
was exponential in ; moreover, for other values of it gives the first
embeddings whose distortion is independent of the graph size . Furthermore,
we use the fact that planar graphs have SPD depth to give a new
proof that any planar graph embeds into with distortion . Our approach also gives new results for graphs with bounded treewidth,
and for graphs excluding a fixed minor
Scales for co-compact embeddings of virtually free groups
Let be a group which is virtually free of rank at least 2 and let
be the family of totally disconnected, locally
compact groups containing as a co-compact lattice.
We prove that the values of the scale function with respect to groups in
evaluated on the subset have only finitely
many prime divisors. This can be thought of as a uniform property of the family
.Comment: 12 pages; key words: uniform lattice, virtually free group, totally
disconnected group, scale function (Error in references corrected in version
2
Minimum cycle and homology bases of surface embedded graphs
We study the problems of finding a minimum cycle basis (a minimum weight set
of cycles that form a basis for the cycle space) and a minimum homology basis
(a minimum weight set of cycles that generates the -dimensional
()-homology classes) of an undirected graph embedded on a
surface. The problems are closely related, because the minimum cycle basis of a
graph contains its minimum homology basis, and the minimum homology basis of
the -skeleton of any graph is exactly its minimum cycle basis.
For the minimum cycle basis problem, we give a deterministic
-time algorithm for graphs embedded on an orientable
surface of genus . The best known existing algorithms for surface embedded
graphs are those for general graphs: an time Monte Carlo
algorithm and a deterministic time algorithm. For the
minimum homology basis problem, we give a deterministic -time algorithm for graphs embedded on an orientable or non-orientable
surface of genus with boundary components, assuming shortest paths are
unique, improving on existing algorithms for many values of and . The
assumption of unique shortest paths can be avoided with high probability using
randomization or deterministically by increasing the running time of the
homology basis algorithm by a factor of .Comment: A preliminary version of this work was presented at the 32nd Annual
International Symposium on Computational Geometr
Fundamental Cycles and Graph Embeddings
In this paper we present a new Good Characterization of maximum genus of a
graph which makes a common generalization of the works of Xuong, Liu, and Fu et
al. Based on this, we find a new polynomially bounded algorithm to find the
maximum genus of a graph
On the homology of the space of knots
Consider the space of `long knots' in R^n, K_{n,1}. This is the space of
knots as studied by V. Vassiliev. Based on previous work of the authors, it
follows that the rational homology of K_{3,1} is free Gerstenhaber-Poisson
algebra. A partial description of a basis is given here. In addition, the mod-p
homology of this space is a `free, restricted Gerstenhaber-Poisson algebra'.
Recursive application of this theorem allows us to deduce that there is
p-torsion of all orders in the integral homology of K_{3,1}.
This leads to some natural questions about the homotopy type of the space of
long knots in R^n for n>3, as well as consequences for the space of smooth
embeddings of S^1 in S^3.Comment: 36 pages, 6 figures. v3: small revisions before publicatio
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