## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Tuesday, April 03, 2007, 12:15 pm

Duration: This information is not available in the database

Location: CAB G51

Speaker: Micha Sharir (Tel Aviv Univ.)

## Weak epsilon nets in convex position: (inverse) Ackermann strikes again

Let S be a set of n points in convex position in the plane, and eps=1/r a parameter. A set N is called weak eps-net for convex subsets of S, if every convex sets that contains at least eps*n points of S contains a point of N.

We show the existence of weak (1/r)-nets with O(r\alpha(r)) points, where \alpha(r) is the inverse Ackermann function. We also derive similar bounds for point sets on the moment curve in any dimension.

The proof is based on reducing the problem to that of stabbing interval chains: Given parameters k\ge j, a k-chain in [1,n] is a sequence of k consecutive disjoint and nonempty intervals of [1,n]. The goal is to find a small set of j-tuples that stab every k-interval chain: For each such chain there exists a j-tuple in our collection whose elements fall into j distinct intervals of the chain. We establish upper and lower bounds on the number of j-tuples that stab every k-interval chain in [1,n].

Joint work with Noga Alon, Haim Kaplan, Gabriel Nivasch and Shakhar Smorodinsky.

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