Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Wednesday, July 05, 2006, 12:15 pm

Duration: This information is not available in the database

Location: OAT S15/S16/S17

Speaker: Csaba Dávid Tóth (Massachusetts Institute of Technology)

Spanning Trees and Cuttings across Disks and Axis-Aligned Rectangles in Three-Space

Given $n$ points and a set of disjoint 2-dimensional {\em disk}s (barriers), we connect the points with a straight line spanning tree such that every disk cuts at most $O(\sqrt{n})$ edges of the tree. If the barriers are {\em axis-aligned rectangles}, then there is a straight line spanning tree such that every rectangle cuts only $O(n^{1/3})$ edges. Both bounds are best possible apart from constant factors. Our proofs crucially depend on new asymptotically tight bounds on {\em cuttings}: for disjoint disks in $\R^3$ and $r$, $1\leq r\leq n$, we construct a $\frac{1}{r}$-cutting of size $O(r^2)$. For axis-aligned rectangles in $\R^3$, we construct a $\frac{1}{r}$-cutting of size $O(r^{3/2})$.

Joint work with Eynat Rafalin and Diane Souvaine.

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