# Problem 23: Approximate 2D Width

From Open Problems in Sublinear Algorithms

Suggested by | Pankaj Agarwal and Piotr Indyk |
---|---|

Source | Kanpur 2009 |

Short link | http://sublinear.info/23 |

The width of a set $P$ of points in the plane is defined as \[ \operatorname{width}(P)=\min_{\|a\|_2=1} \left(\max_{p \in P} a \cdot p-\min_{p \in P} a \cdot p\right). \] For a stream of insertions and deletions of points from a $[\Delta] \times [\Delta]$ grid, we would like to maintain an approximate width of the point set. We conjecture that there is an algorithm for this problem that achieves a constant approximation factor and uses $\operatorname{polylog}(\Delta+n)$ space.

## [edit] Update

The conjecture has been resolved (in the positive direction) by Andoni and Nguyen [AndoniN-12].