5 research outputs found
On the Shannon entropy of the number of vertices with zero in-degree in randomly oriented hypergraphs
Suppose that you have colours and mutually independent dice, each of
which has sides. Each dice lands on any of its sides with equal
probability. You may colour the sides of each die in any way you wish, but
there is one restriction: you are not allowed to use the same colour more than
once on the sides of a die. Any other colouring is allowed. Let be the
number of different colours that you see after rolling the dice. How should you
colour the sides of the dice in order to maximize the Shannon entropy of ?
In this article we investigate this question. We show that the entropy of
is at most and that the bound is tight, up to a
constant additive factor, in the case of there being equally many coins and
colours. Our proof employs the differential entropy bound on discrete entropy,
along with a lower bound on the entropy of binomial random variables whose
outcome is conditioned to be an even integer. We conjecture that the entropy is
maximized when the colours are distributed over the sides of the dice as evenly
as possible.Comment: 11 page
On the Shannon entropy of the number of vertices with zero in-degree in randomly oriented hypergraphs
Suppose that you have colours and mutually independent dice, each of which has sides. Each dice lands on any of its sides with equal probability. You may colour the sides of each die in any way you wish, but there is one restriction: you are not allowed to use the same colour more than once on the sides of a die. Any other colouring is allowed. Let be the number of different colours that you see after rolling the dice. How should you colour the sides of the dice in order to maximize the Shannon entropy of ? In this article we investigate this question. It is shown that the entropy of is at most and that the bound is tight, up to a constant additive factor, in the case of there being equally many coins and colours. Our proof employs the differential entropy bound on discrete entropy, along with a lower bound on the entropy of binomial random variables whose outcome is conditioned to be an even integer. We conjecture that the entropy is maximized when the colours are distributed over the sides of the dice as evenly as possible
An Optimal Algorithm for Strict Circular Seriation
We study the problem of circular seriation, where we are given a matrix of
pairwise dissimilarities between objects, and the goal is to find a {\em
circular order} of the objects in a manner that is consistent with their
dissimilarity. This problem is a generalization of the classical {\em linear
seriation} problem where the goal is to find a {\em linear order}, and for
which optimal algorithms are known. Our contributions can be
summarized as follows. First, we introduce {\em circular Robinson matrices} as
the natural class of dissimilarity matrices for the circular seriation problem.
Second, for the case of {\em strict circular Robinson dissimilarity matrices}
we provide an optimal algorithm for the circular seriation
problem. Finally, we propose a statistical model to analyze the well-posedness
of the circular seriation problem for large . In particular, we establish
rates on the distance between any circular ordering found
by solving the circular seriation problem to the underlying order of the model,
in the Kendall-tau metric.Comment: 27 pages, 5 figure