365 research outputs found
Tight bounds on the maximum size of a set of permutations with bounded VC-dimension
The VC-dimension of a family P of n-permutations is the largest integer k
such that the set of restrictions of the permutations in P on some k-tuple of
positions is the set of all k! permutation patterns. Let r_k(n) be the maximum
size of a set of n-permutations with VC-dimension k. Raz showed that r_2(n)
grows exponentially in n. We show that r_3(n)=2^Theta(n log(alpha(n))) and for
every s >= 4, we have almost tight upper and lower bounds of the form 2^{n
poly(alpha(n))}. We also study the maximum number p_k(n) of 1-entries in an n x
n (0,1)-matrix with no (k+1)-tuple of columns containing all (k+1)-permutation
matrices. We determine that p_3(n) = Theta(n alpha(n)) and that p_s(n) can be
bounded by functions of the form n 2^poly(alpha(n)) for every fixed s >= 4. We
also show that for every positive s there is a slowly growing function
zeta_s(m) (of the form 2^poly(alpha(m)) for every fixed s >= 5) satisfying the
following. For all positive integers n and B and every n x n (0,1)-matrix M
with zeta_s(n)Bn 1-entries, the rows of M can be partitioned into s intervals
so that at least B columns contain at least B 1-entries in each of the
intervals.Comment: 22 pages, 4 figures, correction of the bound on r_3 in the abstract
and other minor change
Largest Empty Circle Centered on a Query Line
The Largest Empty Circle problem seeks the largest circle centered within the
convex hull of a set of points in and devoid of points
from . In this paper, we introduce a query version of this well-studied
problem. In our query version, we are required to preprocess so that when
given a query line , we can quickly compute the largest empty circle
centered at some point on and within the convex hull of .
We present solutions for two special cases and the general case; all our
queries run in time. We restrict the query line to be horizontal in
the first special case, which we preprocess in time and
space, where is the slow growing inverse of the Ackermann's
function. When the query line is restricted to pass through a fixed point, the
second special case, our preprocessing takes time and space. We use insights from the two special cases to solve the
general version of the problem with preprocessing time and space in and respectively.Comment: 18 pages, 13 figure
Bounding sequence extremal functions with formations
An -formation is a concatenation of permutations of letters.
If is a sequence with distinct letters, then let be
the maximum length of any -sparse sequence with distinct letters which
has no subsequence isomorphic to . For every sequence define
, the formation width of , to be the minimum for which
there exists such that there is a subsequence isomorphic to in every
-formation. We use to prove upper bounds on
for sequences such that contains an alternation
with the same formation width as .
We generalize Nivasch's bounds on by showing that
and for every and , such that denotes the inverse Ackermann function.
Upper bounds on have been used in other
papers to bound the maximum number of edges in -quasiplanar graphs on
vertices with no pair of edges intersecting in more than points.
If is any sequence of the form such that is a letter,
is a nonempty sequence excluding with no repeated letters and is
obtained from by only moving the first letter of to another place in
, then we show that and . Furthermore we prove that
and for every .Comment: 25 page
Sources of Superlinearity in Davenport-Schinzel Sequences
A generalized Davenport-Schinzel sequence is one over a finite alphabet that
contains no subsequences isomorphic to a fixed forbidden subsequence. One of
the fundamental problems in this area is bounding (asymptotically) the maximum
length of such sequences. Following Klazar, let Ex(\sigma,n) be the maximum
length of a sequence over an alphabet of size n avoiding subsequences
isomorphic to \sigma. It has been proved that for every \sigma, Ex(\sigma,n) is
either linear or very close to linear; in particular it is O(n
2^{\alpha(n)^{O(1)}}), where \alpha is the inverse-Ackermann function and O(1)
depends on \sigma. However, very little is known about the properties of \sigma
that induce superlinearity of \Ex(\sigma,n).
In this paper we exhibit an infinite family of independent superlinear
forbidden subsequences. To be specific, we show that there are 17 prototypical
superlinear forbidden subsequences, some of which can be made arbitrarily long
through a simple padding operation. Perhaps the most novel part of our
constructions is a new succinct code for representing superlinear forbidden
subsequences
Extremal problems for ordered (hyper)graphs: applications of Davenport-Schinzel sequences
We introduce a containment relation of hypergraphs which respects linear
orderings of vertices and investigate associated extremal functions. We extend,
by means of a more generally applicable theorem, the n.log n upper bound on the
ordered graph extremal function of F=({1,3}, {1,5}, {2,3}, {2,4}) due to Z.
Furedi to the n.(log n)^2.(loglog n)^3 upper bound in the hypergraph case. We
use Davenport-Schinzel sequences to derive almost linear upper bounds in terms
of the inverse Ackermann function. We obtain such upper bounds for the extremal
functions of forests consisting of stars whose all centers precede all leaves.Comment: 22 pages, submitted to the European Journal of Combinatoric
Reduction of -Regular Noncrossing Partitions
In this paper, we present a reduction algorithm which transforms -regular
partitions of to -regular partitions of .
We show that this algorithm preserves the noncrossing property. This yields a
simple explanation of an identity due to Simion-Ullman and Klazar in connection
with enumeration problems on noncrossing partitions and RNA secondary
structures. For ordinary noncrossing partitions, the reduction algorithm leads
to a representation of noncrossing partitions in terms of independent arcs and
loops, as well as an identity of Simion and Ullman which expresses the Narayana
numbers in terms of the Catalan numbers
Disjoint edges in topological graphs and the tangled-thrackle conjecture
It is shown that for a constant , every simple topological
graph on vertices has edges if it has no two sets of edges such
that every edge in one set is disjoint from all edges of the other set (i.e.,
the complement of the intersection graph of the edges is -free). As an
application, we settle the \emph{tangled-thrackle} conjecture formulated by
Pach, Radoi\v{c}i\'c, and T\'oth: Every -vertex graph drawn in the plane
such that every pair of edges have precisely one point in common, where this
point is either a common endpoint, a crossing, or a point of tangency, has at
most edges
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