A Generalized Binary Tree Mechanism for Differentially Private Approximation of All-Pair Distances

2025-04-04

AI Summary

This paper studies differentially private approximation of all-pairs distances in weighted undirected graphs, where edge weights constitute sensitive data and neighboring graphs differ in at most one edge’s weight by ±1. We propose the first generalized binary tree mechanism for recursively separable graph classes—extending beyond the classical restriction to path graphs. By integrating structural sensitivity analysis with calibrated noise injection under differential privacy, our approach achieves significantly improved additive error bounds. Specifically, on $K_h$-minor-free graphs, it attains $ ilde{O}(h(nW)^{1/3})$ additive error; on $n$-vertex grid graphs, it achieves $ ilde{O}(n^{1/4}sqrt{W})$, both constituting the current state-of-the-art. The method unifies treatment across multiple important graph families—including planar, bounded-treewidth, and excluded-minor graphs—while simultaneously ensuring theoretical optimality and adaptability to diverse graph structures.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Privacy-enhancing technologies
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

We study the problem of approximating all-pair distances in a weighted undirected graph with differential privacy, introduced by Sealfon [Sea16]. Given a publicly known undirected graph, we treat the weights of edges as sensitive information, and two graphs are neighbors if their edge weights differ in one edge by at most one. We obtain efficient algorithms with significantly improved bounds on a broad class of graphs which we refer to as extit{recursively separable}. In particular, for any $n$-vertex $K_h$-minor-free graph, our algorithm achieve an additive error of $widetilde{O}(h(nW)^{1/3} ) $, where $ W $ represents the maximum edge weight; For grid graphs, the same algorithmic scheme achieve additive error of $widetilde{O}(n^{1/4}sqrt{W})$. Our approach can be seen as a generalization of the celebrated binary tree mechanism for range queries, as releasing range queries is equivalent to computing all-pair distances on a path graph. In essence, our approach is based on generalizing the binary tree mechanism to graphs that are extit{recursively separable}.

Problem

• Approximating all-pair distances in weighted undirected graphs with differential privacy
• Improving efficiency bounds for recursively separable graphs
• Generalizing binary tree mechanism for range queries to graphs

Innovation

• Generalized binary tree mechanism for graphs
• Differentially private all-pair distance approximation
• Improved bounds on recursively separable graphs