We study the top-$k$ selection problem under the differential privacy model: $m$ items are rated according to votes of a set of clients. We consider a setting in which algorithms can retrieve data via a sequence of accesses, each either a random access or a sorted access; the goal is to minimize the total number of data accesses. Our algorithm requires only $O(\sqrt{mk})$ expected accesses: to our knowledge, this is the first sublinear data-access upper bound for this problem. Accompanying this, we develop the first lower bounds for the problem, in three settings: only random accesses; only sorted acceses; a sequence of accesses of either kind. We show that, to avoid $\Omega(m)$ access cost, supporting \emph{either} kind of access, i.e. the freedom to mix, is necessary, and that in this case our algorithm's access cost is almost optimal.

翻译：我们在差异隐私模式下研究最高至k美元的选择问题: $m美元的项目根据一组客户的投票额进行评级。 我们考虑一个设置, 算法可以通过一个访问序列, 每一个随机访问或分类访问来检索数据; 目标是将数据访问的总数最小化。 我们的算法只要求$O (\\ sqrt{mk})$( $) 的预期访问量: 据我们所知, 这是这个问题第一个次线性数据访问量上限。 与此相伴随, 我们开发了问题的第一个下限, 在三个设置中: 仅随机访问; 仅对acces 进行分类; 任何一种访问的顺序 。 我们显示, 要避免$\ Omega (m) $( m) 访问成本, 支持 kemmph} 访问量的种类, 即混合自由是必要的, 而在这种情况下, 我们的算法访问成本几乎是最佳的 。