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On the two-dimensional davenport-schinzel problem

Jacob T. Schwartz, Micha Sharir
1990 Journal of symbolic computation  
where At(q) is 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.  ... 
doi:10.1016/s0747-7171(08)80070-3 fatcat:phea3vnaujeu3dsi2r6laor6f4

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

Keiko Imai, Hiroshi Imai
1998 Optimization Methods and Software  
In this paper, we show a new reduction of 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.  ... 
doi:10.1080/10556789808805714 fatcat:lbnksjjkdjhlrcwjh7opqenf5u

Improved bounds and new techniques for Davenport--Schinzel sequences and their generalizations

Gabriel Nivasch
2010 Journal of the ACM  
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).  ...  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.  ... 
doi:10.1145/1706591.1706597 fatcat:qowjn2agv5c6vcqsy42m5qy5ua

A generalization of the Kővári-Sós-Turán theorem [article]

Jesse Geneson
2020 arXiv   pre-print
If H is 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.  ... 
arXiv:2002.05336v2 fatcat:4d4owbveo5gmnnjki7v5uzqpay

Improved bounds and new techniques for Davenport–Schinzel sequences and their generalizations [chapter]

Gabriel Nivasch
2009 Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms  
We present several new results regarding λ s (n), 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.  ... 
doi:10.1137/1.9781611973068.1 fatcat:3gkl2ocj55b3zjr5nheuktdnr4

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]

Micha Sharir
1988 Theoretical Foundations of Computer Graphics and CAD  
A near-linear bound 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  ... 
doi:10.1007/978-3-642-83539-1_9 fatcat:45ghlitmvbdnxe7krfgkpymkfa

Sharp Bounds on Davenport-Schinzel Sequences of Every Order

Seth Pettie
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.  ... 
doi:10.1145/2794075 fatcat:gwf37iztfnbxdjgelu3ies3d7q

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

Dan Halperin
1994 Journal of Intelligent and Robotic Systems  
In a motionplanning 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.  ... 
doi:10.1007/bf01258293 fatcat:frsnyobyznh75aqirjjhy5p3te

Voronoi Diagrams of Moving Points

Gerhard Albers, Leonidas J. Guibas, Joseph S. B. Mitchell, Thomas Roos
1998 International journal of computational geometry and applications  
of a (n; s)-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.  ... 
doi:10.1142/s0218195998000187 fatcat:4vydqzhdpjdopnznvtwaozitwq

A survey of motion planning and related geometric algorithms

J.T. Schwartz, M. Sharir
1988 Artificial Intelligence  
Research 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.  ... 
doi:10.1016/0004-3702(88)90053-7 fatcat:ovd6hbal6vgzzbe2rynqxvuafa

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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