Department of Computer Science | Institute of Theoretical Computer Science | CADMO

Prof. Emo Welzl and Prof. Bernd Gärtner

Mittagsseminar Talk Information |

**Date and Time**: Tuesday, September 11, 2012, 11:30 am

**Duration**: 30 minutes

**Location**: CAB G51

**Speaker**: Gábor Tardos (Alfréd Rényi Institute of Mathematics)

Assume we are given a set $S$ of $n$ points in the plane and want find a subset $T$ of these points with the property that every half-plane containing at least $\epsilon n$ points of $S$ contains at least one point of $T$. Such a set $T$ is called an $\epsilon$-net for $S$ with respect to half-planes. The well known theorem of Haussler and Welzl states that an $\epsilon$-net of size $O((1/\epsilon)\log(1/\epsilon))$ exists for any set $S$ and not just with respect to half-planes but also with respect to any any bounded VC-dimension collection of ranges. (All reasonable collections such as triangles, pentagons, disks, etc. have bounded VC dimension.)

While the Haussler-Welzl theorem is very general in many specific geometric settings such as the half-planes or even half-spaces in 3D a stronger statement is also true: $O(1/\epsilon)$ size $\epsilon$-nets exist (Matousek, Seidel, Welzl). In other cases intermediate results were shown: Ezra and Sharir showed recently that for axis-aligned rectangles in the plane or axis-aligned boxes in 3D $O((1/\epsilon)\log\log(1/\epsilon))$ size $\epsilon$-nets exist. Based on these and similar results a "general belief" was formed that the Haussler-Welzl bound is never tight in geometric settings and in "reasonable" cases a linear (that is $O(1/\epsilon)$) bound holds.

Contrary to this belief we construct point sets $S$ in the 4 dimensional Euclidean space such that all $\epsilon$-nets with respect to either half-spaces or axis-aligned boxes have size $\Omega((1/\epsilon)\log(1/\epsilon))$ showing that the Haussler-Welzl bound is tight in this case. In another construction we find planar point sets showing that the Ezra-Sharir bound on $\epsilon$-nets with respect to axis-aligned rectangles is also tight.

Upcoming talks | All previous talks | Talks by speaker | Upcoming talks in iCal format (beta version!)

Previous talks by year: 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996

Information for students and suggested topics for student talks

Automatic MiSe System Software Version 1.4803M | admin login