12 research outputs found

    Sources of Superlinearity in Davenport-Schinzel Sequences

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    A generalized Davenport-Schinzel sequence is one over a finite alphabet that contains no subsequences isomorphic to a fixed forbidden subsequence. One of the fundamental problems in this area is bounding (asymptotically) the maximum length of such sequences. Following Klazar, let Ex(\sigma,n) be the maximum length of a sequence over an alphabet of size n avoiding subsequences isomorphic to \sigma. It has been proved that for every \sigma, Ex(\sigma,n) is either linear or very close to linear; in particular it is O(n 2^{\alpha(n)^{O(1)}}), where \alpha is the inverse-Ackermann function and O(1) depends on \sigma. However, very little is known about the properties of \sigma that induce superlinearity of \Ex(\sigma,n). In this paper we exhibit an infinite family of independent superlinear forbidden subsequences. To be specific, we show that there are 17 prototypical superlinear forbidden subsequences, some of which can be made arbitrarily long through a simple padding operation. Perhaps the most novel part of our constructions is a new succinct code for representing superlinear forbidden subsequences

    Finding the Maximal Empty Rectangle Containing a Query Point

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    Let PP be a set of nn points in an axis-parallel rectangle BB in the plane. We present an O(nα(n)log4n)O(n\alpha(n)\log^4 n)-time algorithm to preprocess PP into a data structure of size O(nα(n)log3n)O(n\alpha(n)\log^3 n), such that, given a query point qq, we can find, in O(log4n)O(\log^4 n) time, the largest-area axis-parallel rectangle that is contained in BB, contains qq, and its interior contains no point of PP. This is a significant improvement over the previous solution of Augustine {\em et al.} \cite{qmex}, which uses slightly superquadratic preprocessing and storage

    Algorithms for Minimum-Cost Paths in Time-Dependent Networks with Waiting Policies

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    We study the problem of computing minimum-cost paths through a time-varying network, in which the travel time and travel cost of each arc are known functions of one's departure time along the arc. For some problem instances, the ability to wait at nodes may allow for less costly paths through the network. When waiting is allowed, it is constrained by a (potentially time-varying) waiting policy that describes the length of time one may wait and the cost of waiting at every node. In discrete time, time-dependent shortest path problems with waiting constraints can be optimally solved by straightforward dynamic programming algorithms; however, for some waiting policies these algorithms can be computationally impractical. In this article, we survey several broad classes of waiting policies and show how techniques for speeding up dynamic programming can be effectively applied to obtain practical algorithms for these different problem variants

    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
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