Problem 23: Approximate 2D Width

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Suggested by Pankaj Agarwal and Piotr Indyk
Source Kanpur 2009
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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.


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