In this paper, we give an almost linear time and space algorithms to sample from an exponential mechanism with an $\ell_1$-score function defined over an exponentially large non-convex set. As a direct result, on input an $n$ vertex $m$ edges graph $G$, we present the \textit{first} $\widetilde{O}(m)$ time and $O(m)$ space algorithms for differentially privately outputting an $n$ vertex $O(m)$ edges synthetic graph that approximates all the cuts and the spectrum of $G$. These are the \emph{first} private algorithms for releasing synthetic graphs that nearly match this task's time and space complexity in the non-private setting while achieving the same (or better) utility as the previous works in the more practical sparse regime. Additionally, our algorithms can be extended to private graph analysis under continual observation.

翻译：本文提出了一种几乎线性时间和空间的算法，用于从定义在指数级大规模非凸集上的$\ell_1$评分函数所对应的指数机制中进行采样。作为直接应用，在输入一个具有$n$个顶点和$m$条边的图$G$时，我们提出了\textit{首个}具有$\widetilde{O}(m)$时间复杂度和$O(m)$空间复杂度的差分隐私算法，用于输出一个具有$n$个顶点和$O(m)$条边的合成图，该图能够近似$G$的所有割以及谱特征。这些是发布合成图的\emph{首个}隐私算法，在非隐私设定下几乎匹配了该任务的时间和空间复杂度，同时在更实用的稀疏图场景中实现了与先前工作相同（或更优）的效用。此外，我们的算法可以扩展到持续观测下的隐私图分析。