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On the two-dimensional davenport-schinzel problem

1990
*
Journal of symbolic computation
*

where At(q) is

doi:10.1016/s0747-7171(08)80070-3
fatcat:phea3vnaujeu3dsi2r6laor6f4
*the*(almost linear) maximum length of a (q, r)*Davenport*-*Schinzel*sequence. ... ) If*the*intersection curve of each pair of functions intersects every plane x = const in at most*two*points, then ~r is at most O(nG+2(n)), where*the*constant of proportionality depends*on*s and t, and ... All this gives relatively satisfactory information concerning*the*"*one*-*dimensional*"*Davenport*-*Schinzel**problem*. ...##
###
Page 3374 of Mathematical Reviews Vol. , Issue 92f
[page]

1992
*
Mathematical Reviews
*

This paper generalizes

*the**one*-*dimensional**Davenport*-*Schinzel**problem*to*two*dimensions and provides solutions for various special cases. ... We show that*the**two*models are essentially equal.” 92f:68070 68Q25 68U05 Schwartz, Jacob T. (1-NY-X); Sharir, Micha (IL-TLAV)*On**the**two*-*dimensional**Davenport*-*Schinzel**problem*. J. ...##
###
Dynamic weighted Voronoi diagrams and weighted minimax matching of two corresponding point sets

1998
*
Optimization Methods and Software
*

In this paper, we show a new reduction of

doi:10.1080/10556789808805714
fatcat:lbnksjjkdjhlrcwjh7opqenf5u
*the**problem*to*the**two*-*dimensional**Davenport*-*Schinzel*sequences, and provide a much simpler proof for*the*almost cubic bound. ... A weighted geometric fitting*problem*between*two*corresponding sets of points is to minimize*the*maximum weighted distance between*two*corresponding pairs of points by translating and rotating*one*set ... Acknowledgment*The*authors would like to thank both referees for their valuable comments*on**the*original version of this paper. ...##
###
Improved bounds and new techniques for Davenport--Schinzel sequences and their generalizations

2010
*
Journal of the ACM
*

Let lambda_s(n) denote

doi:10.1145/1706591.1706597
fatcat:qowjn2agv5c6vcqsy42m5qy5ua
*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). ... obtain improved upper bounds for*the*generalized*Davenport*-*Schinzel*sequences considered by Adamec, Klazar, and Valtr (1992). ... Our new technique is based*on*a variant of*Davenport*-*Schinzel*sequences, in which we turn*the**problem*around, in a sense. We call our variant sequences almost-DS sequences. ...##
###
A generalization of the Kővári-Sós-Turán theorem
[article]

2020
*
arXiv
*
pre-print

If H is

arXiv:2002.05336v2
fatcat:4d4owbveo5gmnnjki7v5uzqpay
*the*1-uniform hypergraph*on*s vertices with s edges, then K_H, t = K_s, t. ... We also obtain analogous bounds for 0-1 matrix Turán*problems*. ... It would be interesting to see if*the*letter method is useful for other Turán-type*problems*, and what else can be said about ex d+1 (n, K H,t ) in general. ...##
###
Improved bounds and new techniques for Davenport–Schinzel sequences and their generalizations
[chapter]

2009
*
Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
*

We present several new results regarding λ s (n),

doi:10.1137/1.9781611973068.1
fatcat:3gkl2ocj55b3zjr5nheuktdnr4
*the*maximum length of a*Davenport*-*Schinzel*sequence of order s*on*n distinct symbols. ... bounds for*the*generalized*Davenport*-*Schinzel*sequences considered by Adamec, Klazar, and Valtr (1992) . ... Our new technique is based*on*a variant of*Davenport*-*Schinzel*sequences, in which we turn*the**problem*around, in a sense. We call our variant sequences almost-DS sequences. ...##
###
Page 3936 of Mathematical Reviews Vol. , Issue 94g
[page]

1994
*
Mathematical Reviews
*

Arrangements of curves in

*the*plane; 3. Lower envelopes and*Davenport*-*Schinzel*sequences (with subsections devoted to appli- cations); 4. ... We hope that*the*reader has obtained a feel for some of*the*beautiful mathematical tools used, such as*Davenport*-*Schinzel*sequences, random sampling and ¢ nets, etc., as well as for some of*the*most important ...##
###
Davenport-Schinzel Sequences and their Geometric Applications
[chapter]

1988
*
Theoretical Foundations of Computer Graphics and CAD
*

A near-linear bound

doi:10.1007/978-3-642-83539-1_9
fatcat:45ghlitmvbdnxe7krfgkpymkfa
*on**the*maximum length of*Davenport*-*Schinzel*sequences enable us to derive sharp bounds*on**the*combinatorial structure underlying various geometric*problems*, which in turn yields efficient ... An (n, s)*Davenport*-*Schinzel*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 ... Section 6 presents basic combinatorial geometric applications of*Davenport*-*Schinzel*sequences to*two*-*dimensional*arrangements of lines, segments, and arcs, and studies*the*role that these sequences play ...##
###
Sharp Bounds on Davenport-Schinzel Sequences of Every Order

2015
*
Journal of the ACM
*

*One*of

*the*oldest unresolved

*problems*in extremal combinatorics is to determine

*the*maximum length of

*Davenport*-

*Schinzel*sequences, where an order-s DS sequence is defined to be

*one*over an n-letter alphabet ... These sequences were introduced by

*Davenport*and

*Schinzel*in 1965 to model a certain

*problem*in differential equations and have since become an indispensable tool in computational geometry and

*the*analysis ...

*Davenport*and

*Schinzel*[9] established n 1+o (1) upper bounds

*on*

*the*length of DS sequences of every order s. ...

##
###
Page 7548 of Mathematical Reviews Vol. , Issue 98M
[page]

1998
*
Mathematical Reviews
*

Summary: “

*One*class of*Davenport*-*Schinzel*sequences consists of finite sequences over n symbols without immediate repetitions and without any subsequence of*the*type abub. ... We show that, for each positive integer m,*the*A,,(n,k) are asymptotically normal.” 98m:05004 OSAI5 82B41 Bousquet-Meélou, Mireille (F-BORD-LB; Talence) New enumerative results*on**two*-*dimensional*directed ...##
###
Page 6907 of Mathematical Reviews Vol. , Issue 2003i
[page]

2003
*
Mathematical Reviews
*

Further related

*problems**on**the*real- izability of*Davenport*-*Schinzel*sequences are discussed. ... (IL-TECH-OR; Haifa) Momentopes,*the*complexity of vector partitioning, and*Davenport*-*Schinzel*sequences. (English summary) Discrete Comput. Geom. 27 (2002), no. 3, 409-417. ...##
###
Robot motion planning and the single cell problem in arrangements

1994
*
Journal of Intelligent and Robotic Systems
*

In a motionplanning

doi:10.1007/bf01258293
fatcat:frsnyobyznh75aqirjjhy5p3te
*problem*with*two*degrees of freedom, for instance,*the*points representing semi-free placements of*the*robot lie*on*several curves in a*two*-*dimensional*con guration space. ... (*The*entire discussion applies to*problems*with more degrees of freedom as well, but this survey only deals with*problems*with*two*o r three degrees of freedom.) ... For*two*-*dimensional*instances of*the*single cell*problem*we h a ve used*Davenport*-*Schinzel*sequences. ...##
###
Voronoi Diagrams of Moving Points

1998
*
International journal of computational geometry and applications
*

of a (n; s)-

doi:10.1142/s0218195998000187
fatcat:4vydqzhdpjdopnznvtwaozitwq
*Davenport*-*Schinzel*sequence AgShSh 89, DaSc 65] and s is a constant depending*on**the*motions of*the*point sites. ... Our results are a linear-factor improvement over*the*naive O(n d+2 ) upper bound*on**the*number of topological events. ... Acknowledgement*The*authors wish to thank Peter Widmayer for carefully reading*the*manuscript. ...##
###
A survey of motion planning and related geometric algorithms

1988
*
Artificial Intelligence
*

Research

doi:10.1016/0004-3702(88)90053-7
fatcat:ovd6hbal6vgzzbe2rynqxvuafa
*on*theoretical*problems*in robotics looks ahead to a future generation of robots substantially more autonomous than present robotic systems, whose algorithmic and software capabilities remain ... three-*dimensional*-spaee4^amidst-a=eolleetion=of=obstacle8=wh^ system. ...*Davenport*-*Schinzel*sequences described below. ...##
###
Page 2259 of Mathematical Reviews Vol. , Issue 89D
[page]

1989
*
Mathematical Reviews
*

Summary: “

*Davenport*-*Schinzel*sequences are sequences that do not contain forbidden alternating subsequences of certain length. ... {For*the*entire collection see MR 89b:6801 1.} Peter Gritzmann (D-TRR) 89d:68085 68U05 05A99 68R05 Sharir, Micha (1-NY-X)*Davenport*-*Schinzel*sequences and their geometric applications. ...
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