Differentially Private Minimum Spanning Tree in Euclidean Graphs

Zongrui Zou, Alessandro Epasto, Chenglin Fan, Rudrajit Das

Abstract: We study differentially private approximation of minimum spanning trees (MST) and hierarchical clustering for Euclidean graph embeddings. Our algorithms achieve an optimal trade-off, providing a $(1+\eta)$-multiplicative approximation with $\tilde{O}(n/\eta^2)$ additive error under $\rho$-dist privacy. Furthermore, we establish a separation between Euclidean and general graphs by proving a lower bound of $\Omega(n^{1.5})$ additive error for general graphs under a similar privacy notion, demonstrating that better utility is indeed achievable for geometric data. Our algorithm can also be directly applied to clustering tasks based on specific MST algorithms, incurring only a minimal loss in the approximation guarantee compared to its non-private counterpart.

Submission Number: 190