Oblivious Median Slope Selection

We study the median slope selection problem in the oblivious RAM model. In this model memory accesses have to be independent of the data processed, i.e., an adversary cannot use observed access patterns to derive additional information about the input. We show how to modify the randomized algorithm of Matou\v{s}ek (1991) to obtain an oblivious version with $\mathcal{O}(n \log^2 n)$ expected time for $n$ points in $\mathbb{R}^2$. This complexity matches a theoretical upper bound that can be obtained through general oblivious transformation. In addition, results from a proof-of-concept implementation show that our algorithm is also practically efficient.