8 research outputs found
Eppstein's bound on intersecting triangles revisited
Let S be a set of n points in the plane, and let T be a set of m triangles
with vertices in S. Then there exists a point in the plane contained in
Omega(m^3 / (n^6 log^2 n)) triangles of T. Eppstein (1993) gave a proof of this
claim, but there is a problem with his proof. Here we provide a correct proof
by slightly modifying Eppstein's argument.Comment: Minor revision following referee's suggestions. To appear in Journal
of Combinatorial Theory, Series A. 5 pages, 1 figur
Realizability of hypergraphs and Ramsey link theory
We present short simple proofs of Conway-Gordon-Sachs' theorem on graphs in
3-dimensional space, as well as van Kampen-Flores' and Ummel's theorems on
nonrealizability of certain hypergraphs (or simplicial complexes) in
4-dimensional space. The proofs use a reduction to lower dimensions which
allows to exhibit relation between these results.
We present a simplified exposition accessible to non-specialists in the area
and to students who know basic geometry of 3-dimensional space and who are
ready to learn straightforward 4-dimensional generalizations. We use elementary
language (e.g. collections of points) which allows to present the main ideas
without technicalities (e.g. without using the formal definition of a
hypergraph).Comment: 19 pages, 11 figures; the paper is rewritten; exposition improve
RRR: Rank-Regret Representative
Selecting the best items in a dataset is a common task in data exploration.
However, the concept of "best" lies in the eyes of the beholder: different
users may consider different attributes more important, and hence arrive at
different rankings. Nevertheless, one can remove "dominated" items and create a
"representative" subset of the data set, comprising the "best items" in it. A
Pareto-optimal representative is guaranteed to contain the best item of each
possible ranking, but it can be almost as big as the full data. Representative
can be found if we relax the requirement to include the best item for every
possible user, and instead just limit the users' "regret". Existing work
defines regret as the loss in score by limiting consideration to the
representative instead of the full data set, for any chosen ranking function.
However, the score is often not a meaningful number and users may not
understand its absolute value. Sometimes small ranges in score can include
large fractions of the data set. In contrast, users do understand the notion of
rank ordering. Therefore, alternatively, we consider the position of the items
in the ranked list for defining the regret and propose the {\em rank-regret
representative} as the minimal subset of the data containing at least one of
the top- of any possible ranking function. This problem is NP-complete. We
use the geometric interpretation of items to bound their ranks on ranges of
functions and to utilize combinatorial geometry notions for developing
effective and efficient approximation algorithms for the problem. Experiments
on real datasets demonstrate that we can efficiently find small subsets with
small rank-regrets