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With the popularization of different kinds of smart terminals and thedevelopment of autonomous driving technology, more and more services based onspatio-temporal data have emerged in our lives, such as online taxi services,traffic flow prediction, and tracking virus propagation. However, the privacyconcerns of spatio-temporal data greatly limit the use of them. To address thisissue, differential privacy method based on spatio-temporal data has beenproposed. In differential privacy, a good aggregation query can highly improvethe data utility. But the mainstream aggregation query methods are based onarea partitioning, which is difficult to generate trajectory with high utilityfor they are hard to take time and constraints into account. Motivated by this,we propose an aggregation query based on the relationships betweentrajectories, so it can greatly improve the data utility as compared to theexisting methods. The trajectory synthesis task can be regarded as anoptimization problem of finding trajectories that match the relationshipsbetween trajectories. We adopt gradient descent to find new trajectories thatmeet the conditions, and during the gradient descent, we can easily take theconstraints into account by adding penalty terms which area partitioning basedquery is hard to achieve. We carry out extensive experiments to validate thatthe trajectories generated by our method have higher utility and the theoreticanalysis shows that our method is safe and reliable.

A Privacy-Preserving Trajectory Synthesis Method Based on Vector Translation Invariance Supporting Traffic Constraints