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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… (More)

We present a simple randomized algorithm which solves linear programs with <italic>n</italic> constraints and <italic>d</italic> variables in expected <italic>O</italic>(<italic>nde</italic><supscrpt>(<italic>d</italic> ln(<italic>n</italic>+1))<supscrpt>1/4</supscrpt></supscrpt>) time in the unit cost model (where we count the number of arithmetic… (More)

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… (More)

We show that a set of <i>n</i> points in the plane has at most <i>O</i>(10.05<sup><i>n</i></sup>) perfect matchings with crossing-free straight-line embedding. The expected number of perfect crossing-free matchings of a set of <i>n</i> points drawn i.i.d. from an arbitrary distribution in the plane is at most <i>O</i>(9.24<sup><i>n</i></sup>).Several… (More)

We reexamine the notion of relative (p, ε)-approximations, recently introduced in [CKMS06], and establish upper bounds on their size, in general range spaces of finite VC-dimension, using the sampling theory developed in [LLS01] and in several earlier studies [Pol86, Hau92, Tal94]. We also survey the different notions of sampling, used in computational… (More)

Given a triangulation of a simple polygon P, we present linear-time algorithms for solving a collection of problems concerning shortest paths and visibility within P. These problems include calculation of the collection of all shortest paths inside P from a given source vertex s to all the other vertices of P, calculation of the subpolygon of P consisting… (More)

We consider the problem of computing the shortest path between two points in two- or three-dimensional space bounded by polyhedral surfaces. In the 2-D case the problem is easily solved in time <italic>O</italic>(<italic>n</italic><supscrpt>2</supscrpt> log <italic>n</italic>).In the general 3-D case the problem is quite hard to solve, and is not even… (More)