6,733 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
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
A Geometric Approach to the Problem of Unique Decomposition of Processes
This paper proposes a geometric solution to the problem of prime
decomposability of concurrent processes first explored by R. Milner and F.
Moller in [MM93]. Concurrent programs are given a geometric semantics using
cubical areas, for which a unique factorization theorem is proved. An effective
factorization method which is correct and complete with respect to the
geometric semantics is derived from the factorization theorem. This algorithm
is implemented in the static analyzer ALCOOL.Comment: 15 page
A Note on Flips in Diagonal Rectangulations
Rectangulations are partitions of a square into axis-aligned rectangles. A
number of results provide bijections between combinatorial equivalence classes
of rectangulations and families of pattern-avoiding permutations. Other results
deal with local changes involving a single edge of a rectangulation, referred
to as flips, edge rotations, or edge pivoting. Such operations induce a graph
on equivalence classes of rectangulations, related to so-called flip graphs on
triangulations and other families of geometric partitions. In this note, we
consider a family of flip operations on the equivalence classes of diagonal
rectangulations, and their interpretation as transpositions in the associated
Baxter permutations, avoiding the vincular patterns { 3{14}2, 2{41}3 }. This
complements results from Law and Reading (JCTA, 2012) and provides a complete
characterization of flip operations on diagonal rectangulations, in both
geometric and combinatorial terms
Homology of non--overlapping discs
In this paper we describe the homology and cohomology of some natural
bimodules over the little discs operad, whose components are configurations of
non--overlapping discs. At the end we briefly explain how this algebraic
structure intervenes in the study of spaces of non--equal immersions.Comment: 29 page
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