The all-pairs shortest distances (APSD) with differential privacy (DP) problem takes as input an undirected, weighted graph $G = (V,E, \mathbf{w})$ and outputs a private estimate of the shortest distances in $G$ between all pairs of vertices. In this paper, we present a simple $\widetilde{O}(n^{1/3}/\varepsilon)$-accurate algorithm to solve APSD with $\varepsilon$-DP, which reduces to $\widetilde{O}(n^{1/4}/\varepsilon)$ in the $(\varepsilon, \delta)$-DP setting, where $n = |V|$. Our algorithm greatly improves upon the error of prior algorithms, namely $\widetilde{O}(n^{2/3}/\varepsilon)$ and $\widetilde{O}(\sqrt{n}/\varepsilon)$ in the two respective settings, and is the first to be optimal up to a polylogarithmic factor, based on a lower bound of $\widetilde{\Omega}(n^{1/4})$. In the case where a multiplicative approximation is allowed, we give two different constructions of algorithms with reduced additive error. Our first construction allows a multiplicative approximation of $O(k\log{\log{n}})$ and has additive error $\widetilde{O}(k\cdot n^{1/k}/\varepsilon)$ in the $\varepsilon$-DP case and $\widetilde{O}(\sqrt{k}\cdot n^{1/(2k)}/\varepsilon)$ in the $(\varepsilon, \delta)$-DP case. Our second construction allows multiplicative approximation $2k-1$ and has the same asymptotic additive error as the first construction. Both constructions significantly improve upon the currently best-known additive error of, $\widetilde{O}(k\cdot n^{1/2 + 1/(4k+2)}/\varepsilon)$ and $\widetilde{O}(k\cdot n^{1/3 + 2/(9k+3)}/\varepsilon)$, respectively. Our algorithms are straightforward and work by decomposing a graph into a set of spanning trees, and applying a key observation that we can privately release APSD in trees with $O(\text{polylog}(n))$ error.

翻译：差分隐私（DP）下的全对最短距离（APSD）问题以无向加权图 $G = (V,E, \mathbf{w})$ 作为输入，输出 $G$ 中所有顶点对之间最短距离的隐私估计。本文提出一种简单的 $\widetilde{O}(n^{1/3}/\varepsilon)$ 精度算法，用于求解 $\varepsilon$-DP 下的 APSD 问题，在 $(\varepsilon, \delta)$-DP 设置下误差可降至 $\widetilde{O}(n^{1/4}/\varepsilon)$，其中 $n = |V|$。我们的算法显著改进了先前算法的误差（在两种设置下分别为 $\widetilde{O}(n^{2/3}/\varepsilon)$ 和 $\widetilde{O}(\sqrt{n}/\varepsilon)$），并且基于 $\widetilde{\Omega}(n^{1/4})$ 的下界，首次实现了在多项式对数因子内的最优性。在允许乘法近似的情况下，我们给出了两种具有更低加法误差的算法构造。第一种构造允许 $O(k\log{\log{n}})$ 的乘法近似，在 $\varepsilon$-DP 情况下的加法误差为 $\widetilde{O}(k\cdot n^{1/k}/\varepsilon)$，在 $(\varepsilon, \delta)$-DP 情况下为 $\widetilde{O}(\sqrt{k}\cdot n^{1/(2k)}/\varepsilon)$。第二种构造允许 $2k-1$ 的乘法近似，并具有与第一种构造相同的渐近加法误差。两种构造均显著改进了当前已知的最佳加法误差（分别为 $\widetilde{O}(k\cdot n^{1/2 + 1/(4k+2)}/\varepsilon)$ 和 $\widetilde{O}(k\cdot n^{1/3 + 2/(9k+3)}/\varepsilon)$）。我们的算法思路清晰，通过将图分解为一组生成树，并应用一个关键观察结果实现：我们能够以 $O(\text{polylog}(n))$ 的误差在树结构中以差分隐私方式发布 APSD。