Oblivious Median Slope Selection

TL;DR

This paper presents an efficient, oblivious algorithm for median slope selection in the RAM model, ensuring data access patterns reveal no information, with both theoretical and practical performance guarantees.

Contribution

It adapts Matoušek's randomized median slope selection algorithm to an oblivious setting with optimal expected time complexity.

Findings

Expected time complexity is O(n log^2 n).

The algorithm is practically efficient based on implementation results.

Matches theoretical upper bounds for oblivious median slope selection.

Abstract

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.