Hierarchical Spatial Decompositions under Local Differential Privacy

Abstract

The popularity of smartphones, GPS-enabled devices, social networks, and connected vehicles all contribute to the increasing volume of spatial data. Spatial decompositions assist in handling big spatial data, and they have been commonly used in the Differential Privacy (DP) literature for range query answering, spatial indexing, count-of-counts histograms, data summarization, and visualization. However, their applications under the emerging Local DP (LDP) notion are scarce. In this article, we study the problem of building hierarchical spatial decompositions under LDP, focusing on two methods: quadtrees and kd-trees. We develop two solutions for quadtrees: a baseline solution that is inspired by the centralized DP literature, and a proposed solution that utilizes a single data collection step from users, propagates density estimates to remaining nodes, and performs structural corrections to the quadtree. Since kd-trees rely on node medians which are data-dependent, we observe that it is not feasible to build kd-trees using a single data collection step. We therefore propose an iterative solution that constructs kd-trees in top-down fashion by utilizing a novel algorithm for estimating node medians at each tree depth. We experimentally evaluate our quadtree and kd-tree algorithms using four real-world spatial datasets, multiple utility metrics, varying privacy budgets, and tree parameters. Results demonstrate that our algorithms enable the building of accurate spatial decompositions that provide high utility in practice. Notably, our quadtrees and kd-trees achieve substantially lower errors in answering spatial density queries (up to 10-fold improvement) when compared with a state-of-the-art method.