Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
Lectures on Discrete Geometry
Space-time tradeoff for answering range queries (Extended Abstract)
STOC '82 Proceedings of the fourteenth annual ACM symposium on Theory of computing
Superconcentrators, generalizers and generalized connectors with limited depth
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
New Constructions of Weak ε-Nets
Discrete & Computational Geometry
Weak ε-nets have basis of size o(1/ε log (1/ε)) in any dimension
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Improved bounds and new techniques for Davenport--Schinzel sequences and their generalizations
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Lower bounds for weak epsilon-nets and stair-convexity
Proceedings of the twenty-fifth annual symposium on Computational geometry
Improved bounds and new techniques for Davenport--Schinzel sequences and their generalizations
Journal of the ACM (JACM)
Generalized Davenport-Schinzel sequences and their 0-1 matrix counterparts
Journal of Combinatorial Theory Series A
Origins of Nonlinearity in Davenport-Schinzel Sequences
SIAM Journal on Discrete Mathematics
Piercing quasi-rectangles-On a problem of Danzer and Rogers
Journal of Combinatorial Theory Series A
Sharp bounds on Davenport-Schinzel sequences of every order
Proceedings of the twenty-ninth annual symposium on Computational geometry
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We construct weak ε-nets of almost linear size forcertain types of point sets. Specifically, for planar point sets inconvex position we construct weak 1/r-nets of size O(rα(r)),where α(r) denotes the inverse Ackermann function. For pointsets along the moment curve in ℝd we constructweak 1/r-nets of size r · 2poly(α(r)),where the degree of the polynomial in the exponent depends(quadratically) on d.Our constructions result from a reduction to a new problem,which we call stabbing interval chains with j-tuples. Given therange of integers N = [1, n], an interval chain of length k is asequence of k consecutive, disjoint, nonempty intervals containedin N. A j-tuple $\bar{P}$ = (p1,…,pj) is said to stab aninterval chain C = I1…Ik if eachpi falls on a different interval of C. The problem is toconstruct a small-size family Z of j-tuples that stabs allk-interval chains in N.Let z(j)k(n) denote the minimum size ofsuch a family Z. We derive almost-tight upper and lower bounds forz(j)k(n) for every fixed j; our boundsinvolve functions αm(n) of the inverse Ackermannhierarchy. Specifically, we show that for j = 3 we havez(3)k(n) =Θ(nα$\lfloor$k/2$\rfloor$(n)) for all k ≥6. For each j≥4, we construct a pair of functionsPʹj(m), Qʹj(m), almost equalasymptotically, such that z(j)Pʹj(m)(n)= O(nαm(n)) andz(j)Qʹj(m)(n) =Ω(nαm(n)).