Davenport-Schinzel sequences and their geometric applications

نویسندگان

  • Micha Sharir
  • Pankaj K. Agarwal
چکیده

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, as a (possibly non-contiguous) subsequence, any alternation a b a b of length s + 2 between two distinct symbols a and b. The close relationship between Davenport{Schinzel 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 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 e cient algorithms for these problems. Both authors have been supported by a grant from the U.S.-Israeli Binational Science Foundation. Pankaj Agarwal has also been supported by a National Science Foundation Grant CCR-93{01259, by an Army Research O ce MURI grant DAAH04-96-1-0013, by a Sloan fellowship, and by an NYI award and matching funds from Xerox Corporation. Micha Sharir has also been supported by NSF Grants CCR-9122103 and CCR-93-11127, by a Max-Planck Research Award, and the Israel Science Fund administered by the Israeli Academy of Sciences, and the G.I.F., the German-Israeli Foundation for Scienti c Research and Development. y Department of Computer Science, Box 90129, Duke University, Durham, NC 27708-0129 z School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel, and Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Generalized Davenport–Schinzel sequences: results, problems, and applications

We survey in detail extremal results on Davenport–Schinzel sequences and their generalizations, from the seminal papers of H. Davenport and A. Schinzel in 1965 to present. We discuss geometric and enumerative applications, generalizations to colored trees, and generalizations to hypergraphs. Eleven illustrative examples with proofs are given and nineteen open problems are posed.

متن کامل

Keywords. Davenport{schinzel Sequence; Tree; Extremal Problem 0 Extremal Problems for Colored Trees and Davenport{schinzel Sequences

In the theory of generalized Davenport{Schinzel sequences one estimates the maximum lengths of nite sequences containing no subsequence of a given pattern. Here we investigate a further generalization, in which the class of sequences is extended to the class of colored trees. We determine exactly the extremal functions associated with the properly 2-colored path of four vertices and with the mo...

متن کامل

Nonlinearity of Davenport-Schinzel Sequences and of a Generalized Path Compression Scheme

Davenport-Schinzel 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 Davenport-Schinzel 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 i...

متن کامل

8. Davenport-schinzel Sequences

Definition 18.1 A (n, s)-Davenport-Schinzel sequence is a sequence over an alphabet A of size n in which no two consecutive characters are the same and there is no alternating subsequence of the form .

متن کامل

Tightish Bounds on Davenport-Schinzel Sequences

Let Ψs(n) be the extremal function of order-s Davenport-Schinzel sequences over an n-letter alphabet. Together with existing bounds due to Hart and Sharir (s = 3), Agarwal, Sharir, and Shor (s = 4, lower bounds on s ≥ 6), and Nivasch (upper bounds on even s), we give the following essentially tight bounds on Ψs(n) for all s: Ψs(n) =  n s = 1

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 1995