16 research outputs found
Extremal families for Kruskal-Katona Theorem
Given a set of size and a positive integer , Kruskal--Katona theorem gives the minimum size of the shadow of a family of -sets of in terms of the cardinality of . We give a characterization of the families of -sets satisfying equality in Kruskal--Katona theorem. This answers a question of F\"uredi and Griggs.Peer ReviewedPostprint (published version
A Tight Upper Bound on the Number of Candidate Patterns
In the context of mining for frequent patterns using the standard levelwise
algorithm, the following question arises: given the current level and the
current set of frequent patterns, what is the maximal number of candidate
patterns that can be generated on the next level? We answer this question by
providing a tight upper bound, derived from a combinatorial result from the
sixties by Kruskal and Katona. Our result is useful to reduce the number of
database scans
The Entropy of Backwards Analysis
Backwards analysis, first popularized by Seidel, is often the simplest most
elegant way of analyzing a randomized algorithm. It applies to incremental
algorithms where elements are added incrementally, following some random
permutation, e.g., incremental Delauney triangulation of a pointset, where
points are added one by one, and where we always maintain the Delauney
triangulation of the points added thus far. For backwards analysis, we think of
the permutation as generated backwards, implying that the th point in the
permutation is picked uniformly at random from the points not picked yet in
the backwards direction. Backwards analysis has also been applied elegantly by
Chan to the randomized linear time minimum spanning tree algorithm of Karger,
Klein, and Tarjan.
The question considered in this paper is how much randomness we need in order
to trust the expected bounds obtained using backwards analysis, exactly and
approximately. For the exact case, it turns out that a random permutation works
if and only if it is minwise, that is, for any given subset, each element has
the same chance of being first. Minwise permutations are known to have
entropy, and this is then also what we need for exact backwards
analysis.
However, when it comes to approximation, the two concepts diverge
dramatically. To get backwards analysis to hold within a factor , the
random permutation needs entropy . This contrasts with
minwise permutations, where it is known that a approximation
only needs entropy. Our negative result for
backwards analysis essentially shows that it is as abstract as any analysis
based on full randomness
Minimising the total number of subsets and supersets
Let be a family of subsets of a ground set
with , and let denote the family
of all subsets of that are subsets or supersets of sets in
. Here we determine the minimum value that
can attain as a function of and . This
can be thought of as a `two-sided' Kruskal-Katona style result. It also gives a
solution to the isoperimetric problem on the graph whose vertices are the
subsets of and in which two vertices are adjacent if one is a
subset of the other. This graph is a supergraph of the -dimensional
hypercube and we note some similarities between our results and Harper's
theorem, which solves the isoperimetric problem for hypercubes. In particular,
analogously to Harper's theorem, we show there is a total ordering of the
subsets of such that, for each initial segment
of this ordering, has the minimum possible size.
Our results also answer a question that arises naturally out of work of Gerbner
et al. on cross-Sperner families and allow us to strengthen one of their main
results.Comment: 21 pages, 1 figur
Shadows and intersections: stability and new proofs
We give a short new proof of a version of the Kruskal-Katona theorem due to
Lov\'asz. Our method can be extended to a stability result, describing the
approximate structure of configurations that are close to being extremal, which
answers a question of Mubayi. This in turn leads to another combinatorial proof
of a stability theorem for intersecting families, which was originally obtained
by Friedgut using spectral techniques and then sharpened by Keevash and Mubayi
by means of a purely combinatorial result of Frankl. We also give an algebraic
perspective on these problems, giving yet another proof of intersection
stability that relies on expansion of a certain Cayley graph of the symmetric
group, and an algebraic generalisation of Lov\'asz's theorem that answers a
question of Frankl and Tokushige.Comment: 18 page