# Duality-based approximation algorithms for depth queries and maximum depth

@article{Aiger2020DualitybasedAA, title={Duality-based approximation algorithms for depth queries and maximum depth}, author={Dror Aiger and Haim Kaplan and Micha Sharir}, journal={ArXiv}, year={2020}, volume={abs/2006.12318} }

We design an efficient data structure for computing a suitably defined approximate depth of any query point in the arrangement $\mathcal{A}(S)$ of a collection $S$ of $n$ halfplanes or triangles in the plane or of halfspaces or simplices in higher dimensions. We then use this structure to find a point of an approximate maximum depth in $\mathcal{A}(S)$. Specifically, given an error parameter $\epsilon>0$, we compute, for any query point $q$, an underestimate $d^-(q)$ of the depth of $q$, that…

## One Citation

### Computing Batched Depth Queries and the Depth of a Set of Points

- GeologyCCCG
- 2022

Simplicial depth and Tukey depth are two common measures for expressing the depth of a point q relative to a set P of points in R. We introduce definitions that generalize these notions to express…

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