TraCS: Trajectory Collection in Continuous Space under Local Differential Privacy

Trajectory collection is essential for location-based services, yet it can reveal highly sensitive information about users, such as daily routines and activities, raising serious privacy concerns. Local Differential Privacy (LDP) offers strong privacy guarantees for users even when the data collector is untrusted. However, existing trajectory collection methods under LDP are largely confined to discrete location spaces, where the size of the location space affects both privacy guarantees and trajectory utility. Moreover, many real-world applications, such as flying trajectories or wearable-sensor traces, naturally operate in continuous spaces, making these discrete-space methods inadequate. This paper shifts the focus from discrete to continuous spaces for trajectory collection under LDP. We propose two methods: TraCS-D, which perturbs the direction and distance of locations, and TraCS-C, which perturbs the Cartesian coordinates of locations. Both methods are theoretically and experimentally analyzed for trajectory utility in continuous spaces. TraCS can also be applied to discrete spaces by rounding perturbed locations to any discrete space embedded in the continuous space. In this case, the privacy and utility guarantees of TraCS are independent of the number of locations in the space, and each perturbation requires only $\Theta(1)$ time complexity. Evaluation results on discrete location spaces validate the efficiency advantage and demonstrate that TraCS outperforms state-of-the-art methods with improved trajectory utility, particularly for large privacy parameters.