Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Tuesday, September 16, 2014, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Luis Barba (Carleton University / Université Libre de Bruxelles)

Detecting intersections between convex polyhedra

In this talk, we revisit the classic problem of preprocessing polyhedra independently so that given two preprocessed polyhedra $P$ and $Q$ in $R^d$ each translated and rotated, their intersection can be tested rapidly. For $d=3$ we show how to perform such a test in $O(\log |P| + \log |Q|)$ time after linear preprocessing time and space. This running time is the best possible and improves upon the last best known query time of $O(\log|P| \log|Q|)$ by Dobkin and Kirkpatrick (1990).

We then generalize this method to any constant dimension $d$, achieving the same optimal $O(\log |P| + \log |Q|)$ query time using a representation of size $O(|P|^{\lfloor d/2 \rfloor +\epsilon})$ for any $\epsilon > 0$ arbitrarily small. This answers an even older question posed by Dobkin and Kirkpatrick 30 years ago.

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