61 research outputs found
On reconstructing n-point configurations from the distribution of distances or areas
One way to characterize configurations of points up to congruence is by
considering the distribution of all mutual distances between points. This paper
deals with the question if point configurations are uniquely determined by this
distribution. After giving some counterexamples, we prove that this is the case
for the vast majority of configurations. In the second part of the paper, the
distribution of areas of sub-triangles is used for characterizing point
configurations. Again it turns out that most configurations are reconstructible
from the distribution of areas, though there are counterexamples.Comment: 21 pages, late
Lossless Representation of Graphs using Distributions
We consider complete graphs with edge weights and/or node weights taking
values in some set. In the first part of this paper, we show that a large
number of graphs are completely determined, up to isomorphism, by the
distribution of their sub-triangles. In the second part, we propose graph
representations in terms of one-dimensional distributions (e.g., distribution
of the node weights, sum of adjacent weights, etc.). For the case when the
weights of the graph are real-valued vectors, we show that all graphs, except
for a set of measure zero, are uniquely determined, up to isomorphism, from
these distributions. The motivating application for this paper is the problem
of browsing through large sets of graphs.Comment: 19 page
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