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

Date and Time: Thursday, December 05, 2019, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Johannes Obenaus

Spanning Trees with Low (Shallow) Stabbing Number

In this talk, I am presenting parts of my Master Thesis, which was supervised by Prof. Welzl, Prof. Mulzer, and Dr. Hoffmann.

For k \in \N and a point set P, a line l is called k-shallow if one of the halfplanes bounded by l contains at most k points of P. The (k-shallow) stabbing number of a geometric graph G (realized on some point set) is the maximum number of intersections any (k-shallow) line determines with the edges of G. Chazelle and Welzl (1989) showed that any n-point set admits a spanning tree of stabbing number O(sqrt(n)), which is tight. Har-Peled and Sharir (2009) proved an upper bound of O(sqrt(k)*log(n/k)) in the shallow setting.
For small k, namely k \leq log n, we prove a lower bound of Omega(k), still leaving a gap to the upper bound but implying that it cannot be improved to O(sqrt(k)) in general.

In the second part, we consider the tree stabbing number of point sets, which is the minimum stabbing number over all spanning trees. We prove that the tree stabbing number is not a monotone parameter, answering a question by Eppstein (2018).

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