Results 1  10
of
69
DavenportSchinzel Sequences and Their Geometric Applications
, 1998
"... An (n; s) DavenportSchinzel sequence, for positive integers n and s, is a sequence composed of n distinct symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly noncontiguous) subsequence, any alternation a \Delta \Delta \Delta b \Delta \ ..."
Abstract

Cited by 479 (121 self)
 Add to MetaCart
(Show Context)
An (n; s) DavenportSchinzel sequence, for positive integers n and s, is a sequence composed of n distinct symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly noncontiguous) subsequence, any alternation a \Delta \Delta \Delta b \Delta \Delta \Delta a \Delta \Delta \Delta b \Delta \Delta \Delta of length s + 2 between two distinct symbols a and b. The close relationship between DavenportSchinzel sequences and the combinatorial structure of lower envelopes of collections of functions make the sequences very attractive because a variety of geometric problems can be formulated in terms of lower envelopes. A nearlinear bound on the maximum length of DavenportSchinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems.
Nonlinearity of DavenportSchinzel sequences and of generalized path compression schemes
 Combinatorica
, 1986
"... DavenportSchinzel sequences are sequences that do not contain forbidden subsequences of alternating symbols. They arise in the computation of the envelope of a set of functions. We show that the maximal length of a DavenportSchinzel sequence composed of n symbols is 6(noc(n»), where t1.(n)is the f ..."
Abstract

Cited by 116 (17 self)
 Add to MetaCart
(Show Context)
DavenportSchinzel sequences are sequences that do not contain forbidden subsequences of alternating symbols. They arise in the computation of the envelope of a set of functions. We show that the maximal length of a DavenportSchinzel sequence composed of n symbols is 6(noc(n»), where t1.(n)is the functional inverse of Ackermann's function, and is thus very slowly increasing to infinity. This is achieved by establishing an equivalence between such sequences and generalized path compression schemes on rooted trees, and then by analyzing these schemes. 1.
Crossings and nestings of matchings and partitions
 Trans. Amer. Math. Soc
"... Abstract. We present results on the enumeration of crossings and nestings for matchings and set partitions. Using a bijection between partitions and vacillating tableaux, we show that if we fix the sets of minimal block elements and maximal block elements, the crossing number and the nesting number ..."
Abstract

Cited by 85 (20 self)
 Add to MetaCart
(Show Context)
Abstract. We present results on the enumeration of crossings and nestings for matchings and set partitions. Using a bijection between partitions and vacillating tableaux, we show that if we fix the sets of minimal block elements and maximal block elements, the crossing number and the nesting number of partitions have a symmetric joint distribution. It follows that the crossing numbers and the nesting numbers are distributed symmetrically over all partitions of [n], as well as over all matchings on [2n]. As a corollary, the number of knoncrossing partitions is equal to the number of knonnesting partitions. The same is also true for matchings. An application is given to the enumeration of matchings with no kcrossing (or with no knesting). 1.
Voronoi Diagrams of Moving Points
, 1995
"... Consider a set of n points in ddimensional Euclidean space, d 2, each of which is continuously moving along a given individual trajectory. At each instant in time, the points define a Voronoi diagram. As the points move, the Voronoi diagram changes continuously, but at certain critical instants in ..."
Abstract

Cited by 58 (6 self)
 Add to MetaCart
Consider a set of n points in ddimensional Euclidean space, d 2, each of which is continuously moving along a given individual trajectory. At each instant in time, the points define a Voronoi diagram. As the points move, the Voronoi diagram changes continuously, but at certain critical instants in time, topological events occur that cause a change in the Voronoi diagram. In this paper, we present a method of maintaining the Voronoi diagram over time, while showing that the number of topological events has an upper bound of O(n d s (n)), where s (n) is the maximum length of a (n; s)DavenportSchinzel sequence [AgShSh 89, DaSc 65] and s is a constant depending on the motions of the point sites. Our results are a linearfactor improvement over the naive O(n d+2 ) upper bound on the number of topological events. In addition, we show that if only k points are moving (while leaving the other n \Gamma k points fixed), there is an upper bound of O(kn d\Gamma1 s (n) + (n \Gamma k)...
A Linear Upper Bound in Extremal Theory of Sequences
, 1994
"... An extremal problem considering sequences related to DavenportSchinzel sequences is investigated in this paper. We prove that f(x k ; n) = O(n) where the quantity on the left side is defined as the maximum length m of the sequence u = a 1 a 2 ::a m of integers such that 1) 1 a r n, 2) ..."
Abstract

Cited by 46 (14 self)
 Add to MetaCart
(Show Context)
An extremal problem considering sequences related to DavenportSchinzel sequences is investigated in this paper. We prove that f(x k ; n) = O(n) where the quantity on the left side is defined as the maximum length m of the sequence u = a 1 a 2 ::a m of integers such that 1) 1 a r n, 2) a r = a s ; r 6= s implies jr \Gamma sj k and 3) u contains no subsequence of the type x k (x stands for xx::x itimes).
On ababFree and abbaFree Set Partitions
, 1996
"... These are partitions of [l] = f1; 2; : : : ; lg into n blocks such that no four term subsequence of [l] induces the mentioned pattern and each k consecutive numbers of [l] fall into different blocks. These structures are motivated by DavenportSchinzel sequences. We summarize and extend known enumer ..."
Abstract

Cited by 45 (8 self)
 Add to MetaCart
These are partitions of [l] = f1; 2; : : : ; lg into n blocks such that no four term subsequence of [l] induces the mentioned pattern and each k consecutive numbers of [l] fall into different blocks. These structures are motivated by DavenportSchinzel sequences. We summarize and extend known enumerative results for the pattern p = abab and give an explicit formula for the number p(abab; n; l; k) of such partitions. Our main tool are generating functions. We determine the corresponding generating function for p = abba and k = 1; 2; 3. For k = 2 there is a connection with the number of directed animals. We solve exactly two related extremal problems.
DavenportSchinzel Theory Of Matrices
"... Let C be a configuration of 1's. We define f(n; C) to be the maximal number of 1's in a 01 matrix of size n \Theta n not having C as a subconfiguration. We consider the problem of determining the order of f(n; C) for several forbidden C's. Among others we prove that f(n; i 1 1 1 ..."
Abstract

Cited by 43 (1 self)
 Add to MetaCart
Let C be a configuration of 1's. We define f(n; C) to be the maximal number of 1's in a 01 matrix of size n \Theta n not having C as a subconfiguration. We consider the problem of determining the order of f(n; C) for several forbidden C's. Among others we prove that f(n; i 1 1 1 1 j ) = \Theta(ff(n)n), where ff(n) is the inverse of the Ackermann function.
Improved bounds and new techniques for DavenportSchinzel sequences and their generalizations
 In Proceedings 20th ACMSIAM Symposium on Discrete Algorithms (SODA
, 2009
"... We present several new results regarding λs(n), the maximum length of a Davenport–Schinzel sequence of order s on n distinct symbols. First, we prove that λs(n) ≤ n · 2 (1/t!)α(n)t +O(α(n) t−1), n · 2 (1/t!)α(n)t log 2 α(n)+O(α(n) t), s ≥ 4 even; s ≥ 3 odd; where t = ⌊(s − 2)/2⌋, and α(n) denotes th ..."
Abstract

Cited by 31 (2 self)
 Add to MetaCart
(Show Context)
We present several new results regarding λs(n), the maximum length of a Davenport–Schinzel sequence of order s on n distinct symbols. First, we prove that λs(n) ≤ n · 2 (1/t!)α(n)t +O(α(n) t−1), n · 2 (1/t!)α(n)t log 2 α(n)+O(α(n) t), s ≥ 4 even; s ≥ 3 odd; where t = ⌊(s − 2)/2⌋, and α(n) denotes the inverse Ackermann function. The previous upper bounds, by Agarwal, Sharir, and Shor (1989), had a leading coefficient of 1 instead of 1/t! in the exponent. The bounds for even s are now tight up to lowerorder terms in the exponent. These new bounds result from a small improvement on the technique of Agarwal et al. More importantly, we also present a new technique for deriving upper bounds for λs(n). This new technique is based on some recurrences very similar to those used by the author, together with Alon, Kaplan, Sharir, and Smorodinsky (SODA 2008), for the problem of stabbing interval chains with jtuples. With this new technique we: (1) rederive the upper bound of λ3(n) ≤ 2nα(n)+O ( n √ α(n) ) (first shown by Klazar, 1999); (2) rederive 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 λ3(n) ≥ 2nα(n) − O(n) (the previous lower bound (Sharir and Agarwal, 1995) had a coefficient of 1 2), so the coefficient 2 is tight. We also present a simpler variant of the construction of Agarwal, Sharir, and Shor that achieves the known lower bounds of λs(n) ≥ n·2 (1/t!)α(n)t−O(α(n) t−1) for s ≥ 4 even.
On Geometric Graphs With No K Pairwise Parallel Edges
 Discrete Comput. Geom
, 1997
"... A geometric graph is a graph G = (V; E) drawn in the plane so that the vertex set V consists of points in general position and the edge set E consists of straight line segments between points of V . Two edges of a geometric graph are said to be parallel , if they are opposite sides of a convex quadr ..."
Abstract

Cited by 29 (1 self)
 Add to MetaCart
(Show Context)
A geometric graph is a graph G = (V; E) drawn in the plane so that the vertex set V consists of points in general position and the edge set E consists of straight line segments between points of V . Two edges of a geometric graph are said to be parallel , if they are opposite sides of a convex quadrilateral. In this paper we show that, for any fixed k 3, any geometric graph on n vertices with no k pairwise parallel edges contains at most O(n) edges, and any geometric graph on n vertices with no k pairwise crossing edges contains at most O(n log n) edges. We also prove a conjecture of Kupitz that any geometric graph on n vertices with no pair of parallel edges contains at most 2n \Gamma 2 edges. 1 Introduction A geometric graph is a graph G = (V; E) drawn in the plane so that the vertex set V consists of points in general position and the edge set E consists of straight line segments between points of V . See [9] for a survey of results about geometric graphs. Two edges of a geomet...
On Growth Rates of Closed Permutation Classes
, 2003
"... A class of permutations is called closed if 2 implies 2 , where the relation is the natural containment of permutations. Let n be the set of all permutations of 1; 2; : : : ; n belonging to . We investigate the counting functions n 7! j n j of closed classes. Our main result says that if ..."
Abstract

Cited by 25 (0 self)
 Add to MetaCart
A class of permutations is called closed if 2 implies 2 , where the relation is the natural containment of permutations. Let n be the set of all permutations of 1; 2; : : : ; n belonging to . We investigate the counting functions n 7! j n j of closed classes. Our main result says that if j n j < 2 for at least one n 1, then there is a unique k 1 such that F n;k j n j F n;k n holds for all n 1 with a constant c > 0. Here F n;k are the generalized Fibonacci numbers which grow like powers of the largest positive root of x 1. We characterize also the constant and the polynomial growth of closed permutation classes and give two more results on these.