111 research outputs found
Sharp Bounds on Davenport-Schinzel Sequences of Every Order
One of the longest-standing open problems in computational geometry is to
bound the lower envelope of univariate functions, each pair of which
crosses at most times, for some fixed . This problem is known to be
equivalent to bounding the length of an order- Davenport-Schinzel sequence,
namely a sequence over an -letter alphabet that avoids alternating
subsequences of the form with length
. These sequences were introduced by Davenport and Schinzel in 1965 to
model a certain problem in differential equations and have since been applied
to bounding the running times of geometric algorithms, data structures, and the
combinatorial complexity of geometric arrangements.
Let be the maximum length of an order- DS sequence over
letters. What is asymptotically? This question has been answered
satisfactorily (by Hart and Sharir, Agarwal, Sharir, and Shor, Klazar, and
Nivasch) when is even or . However, since the work of Agarwal,
Sharir, and Shor in the mid-1980s there has been a persistent gap in our
understanding of the odd orders.
In this work we effectively close the problem by establishing sharp bounds on
Davenport-Schinzel sequences of every order . Our results reveal that,
contrary to one's intuition, behaves essentially like
when is odd. This refutes conjectures due to Alon et al.
(2008) and Nivasch (2010).Comment: A 10-page extended abstract will appear in the Proceedings of the
Symposium on Computational Geometry, 201
Improved bounds and new techniques for Davenport-Schinzel sequences and their generalizations
Let lambda_s(n) denote the maximum length of a Davenport-Schinzel sequence of
order s on n symbols. For s=3 it is known that lambda_3(n) = Theta(n alpha(n))
(Hart and Sharir, 1986). For general s>=4 there are almost-tight upper and
lower bounds, both of the form n * 2^poly(alpha(n)) (Agarwal, Sharir, and Shor,
1989). Our first result is an improvement of the upper-bound technique of
Agarwal et al. We obtain improved upper bounds for s>=6, which are tight for
even s up to lower-order terms in the exponent. More importantly, we also
present a new technique for deriving upper bounds for lambda_s(n). With this
new technique we: (1) re-derive the upper bound of lambda_3(n) <= 2n alpha(n) +
O(n sqrt alpha(n)) (first shown by Klazar, 1999); (2) re-derive our own new
upper bounds for general s; and (3) obtain improved upper bounds for the
generalized Davenport-Schinzel sequences considered by Adamec, Klazar, and
Valtr (1992). Regarding lower bounds, we show that lambda_3(n) >= 2n alpha(n) -
O(n), and therefore, the coefficient 2 is tight. We also present a simpler
version of the construction of Agarwal, Sharir, and Shor that achieves the
known lower bounds for even s>=4.Comment: To appear in Journal of the ACM. 48 pages, 3 figure
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
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
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