111 research outputs found

    Sharp Bounds on Davenport-Schinzel Sequences of Every Order

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    One of the longest-standing open problems in computational geometry is to bound the lower envelope of nn univariate functions, each pair of which crosses at most ss times, for some fixed ss. This problem is known to be equivalent to bounding the length of an order-ss Davenport-Schinzel sequence, namely a sequence over an nn-letter alphabet that avoids alternating subsequences of the form ababa \cdots b \cdots a \cdots b \cdots with length s+2s+2. 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 λs(n)\lambda_s(n) be the maximum length of an order-ss DS sequence over nn letters. What is λs\lambda_s asymptotically? This question has been answered satisfactorily (by Hart and Sharir, Agarwal, Sharir, and Shor, Klazar, and Nivasch) when ss is even or s3s\le 3. 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 ss. Our results reveal that, contrary to one's intuition, λs(n)\lambda_s(n) behaves essentially like λs1(n)\lambda_{s-1}(n) when ss 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

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

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    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). For general s>=4 there are almost-tight upper and lower bounds, both of the form n * 2^poly(alpha(n)) (Agarwal, Sharir, and Shor, 1989). Our first result is an improvement of the upper-bound technique of Agarwal et al. We obtain improved upper bounds for s>=6, which are tight for even s up to lower-order terms in the exponent. More importantly, we also present a new technique for deriving upper bounds for lambda_s(n). With this new technique we: (1) re-derive the upper bound of lambda_3(n) <= 2n alpha(n) + O(n sqrt alpha(n)) (first shown by Klazar, 1999); (2) re-derive 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 lambda_3(n) >= 2n alpha(n) - O(n), and therefore, the coefficient 2 is tight. We also present a simpler version of the construction of Agarwal, Sharir, and Shor that achieves the known lower bounds for even s>=4.Comment: To appear in Journal of the ACM. 48 pages, 3 figure

    Tight bounds on the maximum size of a set of permutations with bounded VC-dimension

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    The VC-dimension of a family P of n-permutations is the largest integer k such that the set of restrictions of the permutations in P on some k-tuple of positions is the set of all k! permutation patterns. Let r_k(n) be the maximum size of a set of n-permutations with VC-dimension k. Raz showed that r_2(n) grows exponentially in n. We show that r_3(n)=2^Theta(n log(alpha(n))) and for every s >= 4, we have almost tight upper and lower bounds of the form 2^{n poly(alpha(n))}. We also study the maximum number p_k(n) of 1-entries in an n x n (0,1)-matrix with no (k+1)-tuple of columns containing all (k+1)-permutation matrices. We determine that p_3(n) = Theta(n alpha(n)) and that p_s(n) can be bounded by functions of the form n 2^poly(alpha(n)) for every fixed s >= 4. We also show that for every positive s there is a slowly growing function zeta_s(m) (of the form 2^poly(alpha(m)) for every fixed s >= 5) satisfying the following. For all positive integers n and B and every n x n (0,1)-matrix M with zeta_s(n)Bn 1-entries, the rows of M can be partitioned into s intervals so that at least B columns contain at least B 1-entries in each of the intervals.Comment: 22 pages, 4 figures, correction of the bound on r_3 in the abstract and other minor change

    Extremal problems for ordered (hyper)graphs: applications of Davenport-Schinzel sequences

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    We introduce a containment relation of hypergraphs which respects linear orderings of vertices and investigate associated extremal functions. We extend, by means of a more generally applicable theorem, the n.log n upper bound on the ordered graph extremal function of F=({1,3}, {1,5}, {2,3}, {2,4}) due to Z. Furedi to the n.(log n)^2.(loglog n)^3 upper bound in the hypergraph case. We use Davenport-Schinzel sequences to derive almost linear upper bounds in terms of the inverse Ackermann function. We obtain such upper bounds for the extremal functions of forests consisting of stars whose all centers precede all leaves.Comment: 22 pages, submitted to the European Journal of Combinatoric
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