16 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
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
On the Extremal Functions of Acyclic Forbidden 0--1 Matrices
The extremal theory of forbidden 0--1 matrices studies the asymptotic growth
of the function , which is the maximum weight of a matrix
whose submatrices avoid a fixed pattern
. This theory has been wildly successful at resolving
problems in combinatorics, discrete and computational geometry, structural
graph theory, and the analysis of data structures, particularly corollaries of
the dynamic optimality conjecture.
All these applications use acyclic patterns, meaning that when is
regarded as the adjacency matrix of a bipartite graph, the graph is acyclic.
The biggest open problem in this area is to bound for
acyclic . Prior results have only ruled out the strict bound
conjectured by Furedi and Hajnal. It is consistent with prior results that
, and also consistent that
.
In this paper we establish a stronger lower bound on the extremal functions
of acyclic . Specifically, we give a new construction of relatively dense
0--1 matrices with 1s that avoid an acyclic
. Pach and Tardos have conjectured that this type of result is the best
possible, i.e., no acyclic exists for which
On the Complexity of Randomly Weighted Voronoi Diagrams
In this paper, we provide an bound on the expected
complexity of the randomly weighted Voronoi diagram of a set of sites in
the plane, where the sites can be either points, interior-disjoint convex sets,
or other more general objects. Here the randomness is on the weight of the
sites, not their location. This compares favorably with the worst case
complexity of these diagrams, which is quadratic. As a consequence we get an
alternative proof to that of Agarwal etal [AHKS13] of the near linear
complexity of the union of randomly expanded disjoint segments or convex sets
(with an improved bound on the latter). The technique we develop is elegant and
should be applicable to other problems
Sorting Pattern-Avoiding Permutations via 0-1 Matrices Forbidding Product Patterns
We consider the problem of comparison-sorting an -permutation that
avoids some -permutation . Chalermsook, Goswami, Kozma, Mehlhorn, and
Saranurak prove that when is sorted by inserting the elements into the
GreedyFuture binary search tree, the running time is linear in the extremal
function . This is the maximum number
of 1s in an 0-1 matrix avoiding , where
is the permutation matrix of , the Kronecker
product, and . The
same time bound can be achieved by sorting with Kozma and Saranurak's
SmoothHeap.
In this paper we give nearly tight upper and lower bounds on the density of
-free matrices in terms of the inverse-Ackermann
function . \mathrm{Ex}(P_\pi\otimes \text{hat},n) =
\left\{\begin{array}{ll} \Omega(n\cdot 2^{\alpha(n)}), & \mbox{for most
$\pi$,}\\ O(n\cdot 2^{O(k^2)+(1+o(1))\alpha(n)}), & \mbox{for all $\pi$.}
\end{array}\right. As a consequence, sorting -free sequences can be
performed in time. For many corollaries of the
dynamic optimality conjecture, the best analysis uses forbidden 0-1 matrix
theory. Our analysis may be useful in analyzing other classes of access
sequences on binary search trees