Faster Private Minimum Spanning Trees

TL;DR

This paper introduces a faster differentially private algorithm for computing approximate minimum spanning trees in graphs with private edge weights, achieving improved utility and runtime over previous methods.

Contribution

The paper presents a novel private MST algorithm with improved runtime of O(m + n^{3/2} log n) that matches the utility of existing in-place algorithms, using a new simulation of Report-Noisy-Max.

Findings

The new algorithm runs in sublinear time for dense graphs.

It achieves comparable utility to existing in-place algorithms.

Experimental results show significant improvements in utility or runtime.

Abstract

Motivated by applications in clustering and synthetic data generation, we consider the problem of releasing a minimum spanning tree (MST) under edge-weight differential privacy constraints where a graph topology $G=(V,E)$ with $n$ vertices and $m$ edges is public, the weight matrix $\vec{W}\in \mathbb{R}^{n \times n}$ is private, and we wish to release an approximate MST under $\rho$-zero-concentrated differential privacy. Weight matrices are considered neighboring if they differ by at most $\Delta_\infty$ in each entry, i.e., we consider an $\ell_\infty$ neighboring relationship. Existing private MST algorithms either add noise to each entry in $\vec{W}$ and estimate the MST by post-processing or add noise to weights in-place during the execution of a specific MST algorithm. Using the post-processing approach with an efficient MST algorithm takes $O(n^2)$ time on dense graphs but results in an additive error on the weight of the MST of magnitude $O(n^2\log n)$. In-place algorithms give asymptotically better utility, but the running time of existing in-place algorithms is $O(n^3)$ for dense graphs. Our main result is a new differentially private MST algorithm that matches the utility of existing in-place methods while running in time $O(m + n^{3/2}\log n)$ for fixed privacy parameter $\rho$. The technical core of our algorithm is an efficient sublinear time simulation of Report-Noisy-Max that works by discretizing all edge weights to a multiple of $\Delta_\infty$ and forming groups of edges with identical weights. Specifically, we present a data structure that allows us to sample a noisy minimum weight edge among at most $O(n^2)$ cut edges in $O(\sqrt{n} \log n)$ time. Experimental evaluations support our claims that our algorithm significantly improves previous algorithms either in utility or running time.