18,435 research outputs found
Computing the stretch of an embedded graph
Let G be a graph embedded in an orientable surface Σ, possibly with edge weights, and denote by len(γ) the length (the number of edges or the sum of the edge weights) of a cycle γ in G. The stretch of a graph embedded on a surface is the minimum of len(α)· len(β) over all pairs of cycles α and β that cross exactly once. We provide an algorithm to compute the stretch of an
embedded graph in time O(g4n log n) with high probability, or in time O(g4n log2 n) in the worst case, where g is the genus of the surface Σ and n is the
number of vertices in G.Slovenian Research AgencyEuropean Science FoundationCarl-Zeiss-FoundationCzech Science Foundatio
Analysis of Farthest Point Sampling for Approximating Geodesics in a Graph
A standard way to approximate the distance between any two vertices and
on a mesh is to compute, in the associated graph, a shortest path from
to that goes through one of sources, which are well-chosen vertices.
Precomputing the distance between each of the sources to all vertices of
the graph yields an efficient computation of approximate distances between any
two vertices. One standard method for choosing sources, which has been used
extensively and successfully for isometry-invariant surface processing, is the
so-called Farthest Point Sampling (FPS), which starts with a random vertex as
the first source, and iteratively selects the farthest vertex from the already
selected sources.
In this paper, we analyze the stretch factor of
approximate geodesics computed using FPS, which is the maximum, over all pairs
of distinct vertices, of their approximated distance over their geodesic
distance in the graph. We show that can be bounded in terms
of the minimal value of the stretch factor obtained using an
optimal placement of sources as , where is the ratio of the lengths of
the longest and the shortest edges of the graph. This provides some evidence
explaining why farthest point sampling has been used successfully for
isometry-invariant shape processing. Furthermore, we show that it is
NP-complete to find sources that minimize the stretch factor.Comment: 13 pages, 4 figure
Computing the Maximum Slope Invariant in Tubular Groups
We show that the maximum slope invariant for tubular groups is easy to
calculate, and give an example of two tubular groups that are distinguishable
by their maximum slopes but not by edge pattern considerations or isoperimetric
function.Comment: 9 pages, 7 figure
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