71 research outputs found

    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

    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

    On interference among moving sensors and related problems

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    We show that for any set of nn points moving along "simple" trajectories (i.e., each coordinate is described with a polynomial of bounded degree) in d\Re^d and any parameter 2kn2 \le k \le n, one can select a fixed non-empty subset of the points of size O(klogk)O(k \log k), such that the Voronoi diagram of this subset is "balanced" at any given time (i.e., it contains O(n/k)O(n/k) points per cell). We also show that the bound O(klogk)O(k \log k) is near optimal even for the one dimensional case in which points move linearly in time. As applications, we show that one can assign communication radii to the sensors of a network of nn moving sensors so that at any given time their interference is O(nlogn)O(\sqrt{n\log n}). We also show some results in kinetic approximate range counting and kinetic discrepancy. In order to obtain these results, we extend well-known results from ε\varepsilon-net theory to kinetic environments
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