Difference between revisions of "Open Problems:23"
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The width of a set $P$ of points in the plane is defined as | The width of a set $P$ of points in the plane is defined as | ||
| − | \[ \ | + | \[ \operatorname{width}(P)=\min_{\|a\|_2=1} \left(\max_{p \in P} a \cdot p-\min_{p \in P} a |
\cdot p\right). \] | \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. | 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. | ||
Revision as of 15:20, 13 November 2012
| Proposed 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.
Update
The conjecture has been resolved (in the positive direction) by Andoni and Nguyen [AndoniN-12].
